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Graphing y = (cos(x*sqrt(7)/2)+sin(x*sqrt(7)/2))*exp(x/2)+(-1+2*x)*exp(x)/4

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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                                      x                
       /   /    ___\      /    ___\\  -               x
       |   |x*\/ 7 |      |x*\/ 7 ||  2   (-1 + 2*x)*e 
f(x) = |cos|-------| + sin|-------||*e  + -------------
       \   \   2   /      \   2   //            4      
$$f{\left(x \right)} = \frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}$$
f = ((2*x - 1)*exp(x))/4 + (sin((sqrt(7)*x)/2) + cos((sqrt(7)*x)/2))*exp(x/2)
The graph of the function
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
$$\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}} = 0$$
Solve this equation
The points of intersection with the axis X:

Numerical solution
$$x_{1} = -12.4745890596719$$
$$x_{2} = -69.463509085838$$
$$x_{3} = -88.4620756734176$$
$$x_{4} = -100.336179790655$$
$$x_{5} = -3.17190640872514$$
$$x_{6} = -40.9656592185629$$
$$x_{7} = -79.75$$
$$x_{8} = -97.9613589672074$$
$$x_{9} = -52.8397633216579$$
$$x_{10} = -43.3404800233709$$
$$x_{11} = -0.395087551771383$$
$$x_{12} = -67.0886882623905$$
$$x_{13} = -38.5908383374582$$
$$x_{14} = -83.7124340265227$$
$$x_{15} = -45.7153008528249$$
$$x_{16} = -26.7167457533744$$
$$x_{17} = -19.5919728316202$$
$$x_{18} = -36.2160176917213$$
$$x_{19} = -71.8383299092854$$
$$x_{20} = -29.09155127698$$
$$x_{21} = -7.76371933246917$$
$$x_{22} = -14.8401739438527$$
$$x_{23} = -57.5894049685959$$
$$x_{24} = -90.836896496865$$
$$x_{25} = -48.0901216743422$$
$$x_{26} = -59.9642257920497$$
$$x_{27} = -50.4649424984083$$
$$x_{28} = -78.9627923796278$$
$$x_{29} = -74.2131507327329$$
$$x_{30} = -31.4663771661085$$
$$x_{31} = -33.8411963227596$$
$$x_{32} = -21.9671945626422$$
$$x_{33} = -24.3418790569578$$
$$x_{34} = -93.2117173203125$$
$$x_{35} = -76.5879715561803$$
$$x_{36} = -64.7138674389432$$
$$x_{37} = -55.2145841451685$$
$$x_{38} = -62.3390466154951$$
$$x_{39} = -95.5865381437599$$
$$x_{40} = -81.3376132030752$$
$$x_{41} = -17.2183148410628$$
$$x_{42} = -10.074641558904$$
$$x_{43} = -86.0872548499701$$
$$x_{44} = -5.23091656516685$$
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to (cos((x*sqrt(7))/2) + sin((x*sqrt(7))/2))*exp(x/2) + ((-1 + 2*x)*exp(x))/4.
$$\frac{\left(-1 + 0 \cdot 2\right) e^{0}}{4} + \left(\sin{\left(\frac{0 \sqrt{7}}{2} \right)} + \cos{\left(\frac{0 \sqrt{7}}{2} \right)}\right) e^{\frac{0}{2}}$$
The result:
$$f{\left(0 \right)} = \frac{3}{4}$$
The point:
(0, 3/4)
Extrema of the function
In order to find the extrema, we need to solve the equation
$$\frac{d}{d x} f{\left(x \right)} = 0$$
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
$$\frac{d}{d x} f{\left(x \right)} = $$
the first derivative
$$\frac{\left(2 x - 1\right) e^{x}}{4} + \left(- \frac{\sqrt{7} \sin{\left(\frac{\sqrt{7} x}{2} \right)}}{2} + \frac{\sqrt{7} \cos{\left(\frac{\sqrt{7} x}{2} \right)}}{2}\right) e^{\frac{x}{2}} + \frac{\left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}}{2} + \frac{e^{x}}{2} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -94.1259598629357$$
$$x_{2} = -15.755779912503$$
$$x_{3} = -41.8799017533883$$
$$x_{4} = -1.42217576938552$$
$$x_{5} = -68.0029308050137$$
$$x_{6} = -6.20917368947973$$
$$x_{7} = -82.2518557456984$$
$$x_{8} = -77.5022140988035$$
$$x_{9} = -87.0014973925933$$
$$x_{10} = -51.3791850409486$$
$$x_{11} = -89.3763182160408$$
$$x_{12} = -79.877034922251$$
$$x_{13} = -98.8756015098306$$
$$x_{14} = -58.5036475112218$$
$$x_{15} = -49.0043642172247$$
$$x_{16} = -65.6281099815663$$
$$x_{17} = -75.1273932753561$$
$$x_{18} = -56.1288266877833$$
$$x_{19} = -44.2547225685085$$
$$x_{20} = -3.97301510526696$$
$$x_{21} = -30.0057959290573$$
$$x_{22} = -13.3850716616592$$
$$x_{23} = -27.6309819337811$$
$$x_{24} = -106.000063980173$$
$$x_{25} = -72.7525724519086$$
$$x_{26} = -91.7511390394882$$
$$x_{27} = -96.5007806863832$$
$$x_{28} = -46.6295433946397$$
$$x_{29} = -63.2532891581186$$
$$x_{30} = -84.6266765691459$$
$$x_{31} = -25.2561406437611$$
$$x_{32} = -34.7554390930521$$
$$x_{33} = -32.3806190138149$$
$$x_{34} = -10.9991185571585$$
$$x_{35} = -22.8813806247205$$
$$x_{36} = -60.8784683346721$$
$$x_{37} = -20.5063810804705$$
$$x_{38} = -70.3777516284612$$
$$x_{39} = -37.1302601601107$$
$$x_{40} = -8.65277374376613$$
$$x_{41} = -18.1320784532039$$
$$x_{42} = -53.7540058643076$$
$$x_{43} = -39.5050809041852$$
The values of the extrema at the points:
                                                                   /                   ___\                          /                   ___\ 
(-94.1259598629357, -6.26030265779763e-40 + 3.6375377898546e-21*cos\47.0629799314679*\/ 7 / - 3.6375377898546e-21*sin\47.0629799314679*\/ 7 /)

                                                                    /                   ___\                           /                   ___\ 
(-15.75577991250299, -1.16769532367355e-6 + 0.000379031987672711*cos\7.87788995625149*\/ 7 / - 0.000379031987672711*sin\7.87788995625149*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-41.87990175338834, -1.37378799133662e-17 + 8.05183530179873e-10*cos\20.9399508766942*\/ 7 / - 8.05183530179873e-10*sin\20.9399508766942*\/ 7 /)

                                                                /                   ___\                        /                   ___\ 
(-1.4221757693855204, -0.231803513138953 + 0.491109636087551*cos\0.71108788469276*\/ 7 / - 0.491109636087551*sin\0.71108788469276*\/ 7 /)

                                                                   /                   ___\                          /                   ___\ 
(-68.0029308050137, -1.00318621330841e-28 + 1.7113987051587e-15*cos\34.0014654025069*\/ 7 / - 1.7113987051587e-15*sin\34.0014654025069*\/ 7 /)

                                                                 /                   ___\                        /                   ___\ 
(-6.209173689479733, -0.00674573333302558 + 0.044843041871131*cos\3.10458684473987*\/ 7 / - 0.044843041871131*sin\3.10458684473987*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-82.25185574569844, -7.85634722257839e-35 + 1.37795952660143e-18*cos\41.1259278728492*\/ 7 / - 1.37795952660143e-18*sin\41.1259278728492*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-77.50221409880353, -8.55643816730485e-33 + 1.48118068741037e-17*cos\38.7511070494018*\/ 7 / - 1.48118068741037e-17*sin\38.7511070494018*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-87.00149739259334, -7.18977425313771e-37 + 1.28193168672174e-19*cos\43.5007486962967*\/ 7 / - 1.28193168672174e-19*sin\43.5007486962967*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-51.37918504094856, -1.25969846918233e-21 + 6.96869931502913e-12*cos\25.6895925204743*\/ 7 / - 6.96869931502913e-12*sin\25.6895925204743*\/ 7 /)

                                                                    /                   ___\                           /                   ___\ 
(-89.37631821604079, -6.8702645968828e-38 + 3.91002102750183e-20*cos\44.6881591080204*\/ 7 / - 3.91002102750183e-20*sin\44.6881591080204*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-79.87703492225099, -8.20250470185076e-34 + 4.51775058943627e-18*cos\39.9385174611255*\/ 7 / - 4.51775058943627e-18*sin\39.9385174611255*\/ 7 /)

                                                                    /                   ___\                           /                   ___\ 
(-98.8756015098306, -5.69012286610207e-42 + 3.38404348200507e-22*cos\49.4378007549153*\/ 7 / - 3.38404348200507e-22*sin\49.4378007549153*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-58.50364751122175, -1.15355306875257e-24 + 1.97739921463783e-13*cos\29.2518237556109*\/ 7 / - 1.97739921463783e-13*sin\29.2518237556109*\/ 7 /)

                                                                      /                   ___\                           /                   ___\ 
(-49.004364217224676, -1.29207737816685e-20 + 2.28474384295431e-11*cos\24.5021821086123*\/ 7 / - 2.28474384295431e-11*sin\24.5021821086123*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-65.62810998156634, -1.04095053286571e-27 + 5.61095762228959e-15*cos\32.8140549907832*\/ 7 / - 5.61095762228959e-15*sin\32.8140549907832*\/ 7 /)

                                                                     /                  ___\                           /                  ___\ 
(-75.12739327535608, -8.91737019878726e-32 + 4.85616942619051e-17*cos\37.563696637678*\/ 7 / - 4.85616942619051e-17*sin\37.563696637678*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-56.12882668778328, -1.19005726940378e-23 + 6.48306158125984e-13*cos\28.0644133438916*\/ 7 / - 6.48306158125984e-13*sin\28.0644133438916*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-44.25472256850846, -1.34966835085251e-18 + 2.4558910429626e-10*cos\22.1273612842542*\/ 7 / - 2.4558910429626e-10*sin\22.1273612842542*\/ 7 /)

                                                              /                   ___\                       /                   ___\ 
(-3.97301510526696, -0.0420834980941479 + 0.13717366164542*cos\1.98650755263348*\/ 7 / - 0.13717366164542*sin\1.98650755263348*\/ 7 /)

                                                                     /                   ___\                          /                   ___\ 
(-30.005795929057342, -1.41906003086446e-12 + 3.05017109702037e-7*cos\15.0028979645287*\/ 7 / - 3.05017109702037e-7*sin\15.0028979645287*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-13.385071661659248, -1.06771505721615e-5 + 0.00124013401222523*cos\6.69253583082962*\/ 7 / - 0.00124013401222523*sin\6.69253583082962*\/ 7 /)

                                                                     /                   ___\                          /                   ___\ 
(-27.630981933781136, -1.40660420938282e-11 + 1.00001959126561e-6*cos\13.8154909668906*\/ 7 / - 1.00001959126561e-6*sin\13.8154909668906*\/ 7 /)

                                                                     /                   ___\                          /                   ___\ 
(-106.00006398017295, -4.90994927060766e-45 + 9.6023728688568e-24*cos\53.0000319900865*\/ 7 / - 9.6023728688568e-24*sin\53.0000319900865*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-72.75257245190862, -9.28436327157574e-31 + 1.59213401148904e-16*cos\36.3762862259543*\/ 7 / - 1.59213401148904e-16*sin\36.3762862259543*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-91.75113903948824, -6.56037021695846e-39 + 1.19259587650906e-20*cos\45.8755695197441*\/ 7 / - 1.19259587650906e-20*sin\45.8755695197441*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-96.50078068638315, -5.97019728305877e-41 + 1.1094857389036e-21*cos\48.2503903431916*\/ 7 / - 1.1094857389036e-21*sin\48.2503903431916*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-46.629543394639676, -1.3222386464189e-19 + 7.49071553953719e-11*cos\23.3147716973198*\/ 7 / - 7.49071553953719e-11*sin\23.3147716973198*\/ 7 /)

                                                                      /                   ___\                           /                   ___\ 
(-63.253289158118605, -1.07874341219728e-26 + 1.83959736233504e-14*cos\31.6266445790593*\/ 7 / - 1.83959736233504e-14*sin\31.6266445790593*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-84.62667656914589, -7.51860080687726e-36 + 4.20291563104765e-19*cos\42.3133382845729*\/ 7 / - 4.20291563104765e-19*sin\42.3133382845729*\/ 7 /)

                                                                     /                   ___\                          /                   ___\ 
(-25.256140643761082, -1.38435755926957e-10 + 3.27867781545686e-6*cos\12.6280703218805*\/ 7 / - 3.27867781545686e-6*sin\12.6280703218805*\/ 7 /)

                                                                    /                  ___\                          /                  ___\ 
(-34.75543909305208, -1.41938665637297e-14 + 2.83760730100961e-8*cos\17.377719546526*\/ 7 / - 2.83760730100961e-8*sin\17.377719546526*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-32.38061901381489, -1.42293752015255e-13 + 9.30331917520394e-8*cos\16.1903095069074*\/ 7 / - 9.30331917520394e-8*sin\16.1903095069074*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-10.999118557158472, -9.61120987422611e-5 + 0.00408857296313486*cos\5.49955927857924*\/ 7 / - 0.00408857296313486*sin\5.49955927857924*\/ 7 /)

                                                                   /                   ___\                         /                   ___\ 
(-22.881380624720506, -1.35077437281816e-9 + 1.0749080817876e-5*cos\11.4406903123603*\/ 7 / - 1.0749080817876e-5*sin\11.4406903123603*\/ 7 /)

                                                                     /                  ___\                           /                  ___\ 
(-60.87846833467206, -1.11635722260626e-25 + 6.03126717989204e-14*cos\30.439234167336*\/ 7 / - 6.03126717989204e-14*sin\30.439234167336*\/ 7 /)

                                                                   /                   ___\                          /                   ___\ 
(-20.50638108047054, -1.30470731439589e-8 + 3.52448710917598e-5*cos\10.2531905402353*\/ 7 / - 3.52448710917598e-5*sin\10.2531905402353*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-70.37775162846118, -9.65630013227576e-30 + 5.21993877904848e-16*cos\35.1888758142306*\/ 7 / - 5.21993877904848e-16*sin\35.1888758142306*\/ 7 /)

                                                                    /                   ___\                          /                   ___\ 
(-37.13026016011073, -1.40941901370428e-15 + 8.65498760832404e-9*cos\18.5651300800554*\/ 7 / - 8.65498760832404e-9*sin\18.5651300800554*\/ 7 /)

                                                                   /                   ___\                         /                   ___\ 
(-8.652773743766135, -0.000799228277034128 + 0.0132152095789127*cos\4.32638687188307*\/ 7 / - 0.0132152095789127*sin\4.32638687188307*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-18.132078453203864, -1.24328212826405e-7 + 0.000115523194890462*cos\9.06603922660193*\/ 7 / - 0.000115523194890462*sin\9.06603922660193*\/ 7 /)

                                                                     /                   ___\                           /                   ___\ 
(-53.75400586430762, -1.22555748652212e-22 + 2.12552362497459e-12*cos\26.8770029321538*\/ 7 / - 2.12552362497459e-12*sin\26.8770029321538*\/ 7 /)

                                                                   /                   ___\                          /                   ___\ 
(-39.5050809041852, -1.39394780888258e-16 + 2.63985867849902e-9*cos\19.7525404520926*\/ 7 / - 2.63985867849902e-9*sin\19.7525404520926*\/ 7 /)


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
Minima of the function at points:
$$x_{1} = -15.755779912503$$
$$x_{2} = -1.42217576938552$$
$$x_{3} = -68.0029308050137$$
$$x_{4} = -6.20917368947973$$
$$x_{5} = -82.2518557456984$$
$$x_{6} = -77.5022140988035$$
$$x_{7} = -87.0014973925933$$
$$x_{8} = -58.5036475112218$$
$$x_{9} = -49.0043642172247$$
$$x_{10} = -44.2547225685085$$
$$x_{11} = -30.0057959290573$$
$$x_{12} = -106.000063980173$$
$$x_{13} = -72.7525724519086$$
$$x_{14} = -91.7511390394882$$
$$x_{15} = -96.5007806863832$$
$$x_{16} = -63.2532891581186$$
$$x_{17} = -25.2561406437611$$
$$x_{18} = -34.7554390930521$$
$$x_{19} = -10.9991185571585$$
$$x_{20} = -20.5063810804705$$
$$x_{21} = -53.7540058643076$$
$$x_{22} = -39.5050809041852$$
Maxima of the function at points:
$$x_{22} = -94.1259598629357$$
$$x_{22} = -41.8799017533883$$
$$x_{22} = -51.3791850409486$$
$$x_{22} = -89.3763182160408$$
$$x_{22} = -79.877034922251$$
$$x_{22} = -98.8756015098306$$
$$x_{22} = -65.6281099815663$$
$$x_{22} = -75.1273932753561$$
$$x_{22} = -56.1288266877833$$
$$x_{22} = -3.97301510526696$$
$$x_{22} = -13.3850716616592$$
$$x_{22} = -27.6309819337811$$
$$x_{22} = -46.6295433946397$$
$$x_{22} = -84.6266765691459$$
$$x_{22} = -32.3806190138149$$
$$x_{22} = -22.8813806247205$$
$$x_{22} = -60.8784683346721$$
$$x_{22} = -70.3777516284612$$
$$x_{22} = -37.1302601601107$$
$$x_{22} = -8.65277374376613$$
$$x_{22} = -18.1320784532039$$
Decreasing at intervals
$$\left[-1.42217576938552, \infty\right)$$
Increasing at intervals
$$\left(-\infty, -106.000063980173\right]$$
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0$$
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
$$\frac{d^{2}}{d x^{2}} f{\left(x \right)} = $$
the second derivative
$$\frac{\left(2 x - 1\right) e^{x}}{4} - \frac{\sqrt{7} \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} - \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}}{2} - \frac{3 \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}}{2} + e^{x} = 0$$
Solve this equation
The roots of this equation
$$x_{1} = -23.795597977873$$
$$x_{2} = -76.0416358179793$$
$$x_{3} = -33.2948612461761$$
$$x_{4} = -78.4164566414267$$
$$x_{5} = -9.55571058892964$$
$$x_{6} = -97.4150232290064$$
$$x_{7} = -40.4193234575754$$
$$x_{8} = -42.7941442925279$$
$$x_{9} = -54.6682484069427$$
$$x_{10} = -95.0402024055589$$
$$x_{11} = -28.5452216371588$$
$$x_{12} = -52.2934275835347$$
$$x_{13} = -92.6653815821115$$
$$x_{14} = 1.32636860653533$$
$$x_{15} = -68.9171733476369$$
$$x_{16} = -35.6696817373424$$
$$x_{17} = -99.7898440524538$$
$$x_{18} = -85.5409191117691$$
$$x_{19} = -16.6706289046268$$
$$x_{20} = -90.290560758664$$
$$x_{21} = -64.1675317007419$$
$$x_{22} = -61.7927108772949$$
$$x_{23} = -87.9157399352165$$
$$x_{24} = -49.9186067599637$$
$$x_{25} = -14.2976382353054$$
$$x_{26} = -7.15068182582166$$
$$x_{27} = -11.9178781407992$$
$$x_{28} = -4.83674337734319$$
$$x_{29} = -66.5423525241895$$
$$x_{30} = -73.6668149945318$$
$$x_{31} = -21.4206974815977$$
$$x_{32} = -45.1689651122551$$
$$x_{33} = -19.0461075603526$$
$$x_{34} = -83.1660982883216$$
$$x_{35} = -47.5437859369017$$
$$x_{36} = -38.044502669579$$
$$x_{37} = -26.1703916824703$$
$$x_{38} = -71.2919941710844$$
$$x_{39} = -30.9200394132921$$
$$x_{40} = -2.38625998668675$$
$$x_{41} = 0.212071373337196$$
$$x_{42} = -80.7912774648742$$
$$x_{43} = -59.4178900538462$$
$$x_{44} = -57.0430692304027$$

Сonvexity and concavity intervals:
Let’s find the intervals where the function is convex or concave, for this look at the behaviour of the function at the inflection points:
Concave at the intervals
$$\left[1.32636860653533, \infty\right)$$
Convex at the intervals
$$\left(-\infty, -97.4150232290064\right]$$
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
$$\lim_{x \to -\infty}\left(\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}\right) = 0$$
Let's take the limit
so,
equation of the horizontal asymptote on the left:
$$y = 0$$
$$\lim_{x \to \infty}\left(\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}\right) = \infty$$
Let's take the limit
so,
horizontal asymptote on the right doesn’t exist
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of (cos((x*sqrt(7))/2) + sin((x*sqrt(7))/2))*exp(x/2) + ((-1 + 2*x)*exp(x))/4, divided by x at x->+oo and x ->-oo
$$\lim_{x \to -\infty}\left(\frac{\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}}{x}\right) = 0$$
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
$$\lim_{x \to \infty}\left(\frac{\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}}}{x}\right) = \infty$$
Let's take the limit
so,
inclined asymptote on the right doesn’t exist
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
$$\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}} = \frac{\left(- 2 x - 1\right) e^{- x}}{4} + \left(- \sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{- \frac{x}{2}}$$
- No
$$\frac{\left(2 x - 1\right) e^{x}}{4} + \left(\sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{\frac{x}{2}} = - \frac{\left(- 2 x - 1\right) e^{- x}}{4} - \left(- \sin{\left(\frac{\sqrt{7} x}{2} \right)} + \cos{\left(\frac{\sqrt{7} x}{2} \right)}\right) e^{- \frac{x}{2}}$$
- No
so, the function
not is
neither even, nor odd