Mister Exam

Other calculators

  • How to use it?

  • Graphing y =:
  • x^4+8x^3+16x^2
  • x²-3x+1
  • x^1/3 x^1/3
  • 2x^2-6x+4

  • Identical expressions

  • (- three + two *x)*exp(x)/ four
  • ( minus 3 plus 2 multiply by x) multiply by exponent of (x) divide by 4
  • ( minus three plus two multiply by x) multiply by exponent of (x) divide by four
  • (-3+2x)exp(x)/4
  • -3+2xexpx/4
  • (-3+2*x)*exp(x) divide by 4

  • Similar expressions

  • (3+2*x)*exp(x)/4
  • (-3-2*x)*exp(x)/4
Plotting a function graph/ (-3+2*x)*exp(x)/4

Graphing y = (-3+2*x)*exp(x)/4

v

The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

You have entered [src] 
                   x
       (-3 + 2*x)*e 
f(x) = -------------
             4
f(x)=(2x−3)ex4f{\left(x \right)} = \frac{\left(2 x - 3\right) e^{x}}{4} 
f = ((2*x - 3)*exp(x))/4
The graph of the function
02468-8-6-4-2-1010-100000100000
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:
(2x−3)ex4=0\frac{\left(2 x - 3\right) e^{x}}{4} = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=32x_{1} = \frac{3}{2}
Numerical solution
x1=−53.3759496325742x_{1} = -53.3759496325742
x2=−69.2350405585825x_{2} = -69.2350405585825
x3=−101.105362879064x_{3} = -101.105362879064
x4=−55.3525448459029x_{4} = -55.3525448459029
x5=−81.1723542035891x_{5} = -81.1723542035891
x6=−103.100276799833x_{6} = -103.100276799833
x7=−85.1560310834452x_{7} = -85.1560310834452
x8=−39.6404760650587x_{8} = -39.6404760650587
x9=−65.2622468163915x_{9} = -65.2622468163915
x10=−37.7039270635681x_{10} = -37.7039270635681
x11=−87.1485204030551x_{11} = -87.1485204030551
x12=−95.1220841274421x_{12} = -95.1220841274421
x13=−45.4986204818886x_{13} = -45.4986204818886
x14=−71.2228229319478x_{14} = -71.2228229319478
x15=−61.2939118937805x_{15} = -61.2939118937805
x16=−77.1906807996878x_{16} = -77.1906807996878
x17=−113.077853144863x_{17} = -113.077853144863
x18=−35.7794462611513x_{18} = -35.7794462611513
x19=−91.1346357884748x_{19} = -91.1346357884748
x20=−117.070073488174x_{20} = -117.070073488174
x21=−121.062861741813x_{21} = -121.062861741813
x22=−105.095408376221x_{22} = -105.095408376221
x23=−75.200715143261x_{23} = -75.200715143261
x24=−109.086270746787x_{24} = -109.086270746787
x25=−63.2774554894988x_{25} = -63.2774554894988
x26=−73.2114067557342x_{26} = -73.2114067557342
x27=−79.1812445627715x_{27} = -79.1812445627715
x28=−57.3312461733115x_{28} = -57.3312461733115
x29=−47.462553038273x_{29} = -47.462553038273
x30=−49.430492672938x_{30} = -49.430492672938
x31=−43.5395199022792x_{31} = -43.5395199022792
x32=−111.081977383486x_{32} = -111.081977383486
x33=−107.09074388777x_{33} = -107.09074388777
x34=−51.4017952197327x_{34} = -51.4017952197327
x35=−83.1639634344856x_{35} = -83.1639634344856
x36=−89.1413984890304x_{36} = -89.1413984890304
x37=−67.2481479021583x_{37} = -67.2481479021583
x38=−115.073888205425x_{38} = -115.073888205425
x39=−59.3117781587306x_{39} = -59.3117781587306
x40=−97.1162493536177x_{40} = -97.1162493536177
x41=−93.1282056820621x_{41} = -93.1282056820621
x42=−41.5863271755069x_{42} = -41.5863271755069
x43=−33.8710922394818x_{43} = -33.8710922394818
x44=−119.066400594498x_{44} = -119.066400594498
x45=−99.1106815979612x_{45} = -99.1106815979612
x46=−31.985128603532x_{46} = -31.985128603532
x47=1.5x_{47} = 1.5
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to ((-3 + 2*x)*exp(x))/4.
(−3+0⋅2)e04\frac{\left(-3 + 0 \cdot 2\right) e^{0}}{4}
The result:
f(0)=−34f{\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
ddxf(x)=0\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:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
(2x−3)ex4+ex2=0\frac{\left(2 x - 3\right) e^{x}}{4} + \frac{e^{x}}{2} = 0
Solve this equation
The roots of this equation
x1=12x_{1} = \frac{1}{2}
The values of the extrema at the points:
        1/2  
      -e     
(1/2, ------)
        2


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:
x1=12x_{1} = \frac{1}{2}
The function has no maxima
Decreasing at intervals
[12,∞)\left[\frac{1}{2}, \infty\right)
Increasing at intervals
(−∞,12]\left(-\infty, \frac{1}{2}\right]
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\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:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
(2x+1)ex4=0\frac{\left(2 x + 1\right) e^{x}}{4} = 0
Solve this equation
The roots of this equation
x1=−12x_{1} = - \frac{1}{2}

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
[−12,∞)\left[- \frac{1}{2}, \infty\right)
Convex at the intervals
(−∞,−12]\left(-\infty, - \frac{1}{2}\right]
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞((2x−3)ex4)=0\lim_{x \to -\infty}\left(\frac{\left(2 x - 3\right) e^{x}}{4}\right) = 0
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=0y = 0
lim⁡x→∞((2x−3)ex4)=∞\lim_{x \to \infty}\left(\frac{\left(2 x - 3\right) e^{x}}{4}\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 ((-3 + 2*x)*exp(x))/4, divided by x at x->+oo and x ->-oo
lim⁡x→−∞((2x−3)ex4x)=0\lim_{x \to -\infty}\left(\frac{\left(2 x - 3\right) e^{x}}{4 x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞((2x−3)ex4x)=∞\lim_{x \to \infty}\left(\frac{\left(2 x - 3\right) e^{x}}{4 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:
(2x−3)ex4=(−2x−3)e−x4\frac{\left(2 x - 3\right) e^{x}}{4} = \frac{\left(- 2 x - 3\right) e^{- x}}{4}
- No
(2x−3)ex4=−(−2x−3)e−x4\frac{\left(2 x - 3\right) e^{x}}{4} = - \frac{\left(- 2 x - 3\right) e^{- x}}{4}
- No
so, the function
not is
neither even, nor odd
    © Mister Exam – Calculator