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: atan(2x+3)=0 Solve this equation The points of intersection with the axis X:
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0: substitute x = 0 to sqrt(atan(sqrt(2*x + 3))). atan(2⋅0+3) The result: f(0)=33π The point:
(0, sqrt(3)*sqrt(pi)/3)
Extrema of the function
In order to find the extrema, we need to solve the equation dxdf(x)=0 (the derivative equals zero), and the roots of this equation are the extrema of this function: dxdf(x)= the first derivative 22x+3⋅(2x+4)atan(2x+3)1=0 Solve this equation Solutions are not found, function may have no extrema
Inflection points
Let's find the inflection points, we'll need to solve the equation for this dx2d2f(x)=0 (the second derivative equals zero), the roots of this equation will be the inflection points for the specified function graph: dx2d2f(x)= the second derivative −16(x+2)atan(2x+3)(x+2)2x+34+(2x+3)234+(x+2)(2x+3)atan(2x+3)1=0 Solve this equation The roots of this equation x1=107577.56403671 x2=62903.8628749537 x3=60555.049520562 x4=79353.594352919 x5=67602.4245400132 x6=116990.63189526 x7=86407.0907612323 x8=39433.7739186636 x9=105224.644640647 x10=109930.626877423 x11=74652.3743804977 x12=77002.8666831961 x13=69952.1408924897 x14=32403.2572766674 x15=51163.3629419719 x16=37089.5579402752 x17=95814.5013977822 x18=58206.5716651308 x19=102871.873473385 x20=55858.4494176989 x21=46469.998895868 x22=48816.4507038095 x23=98166.7963429332 x24=88758.6644284932 x25=72302.1284767751 x26=114637.164996904 x27=44124.041963804 x28=84055.7155798312 x29=41778.6188611969 x30=93462.376769134 x31=112283.828630407 x32=91110.4288464257 x33=34746.0296041462 x34=81704.5471562711 x35=100519.255590868 x36=65252.9934430143 x37=53510.7049758468 x38=119344.225445893
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: Have no bends at the whole real axis
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo x→−∞limatan(2x+3)=22π Let's take the limit so, equation of the horizontal asymptote on the left: y=22π x→∞limatan(2x+3)=22π Let's take the limit so, equation of the horizontal asymptote on the right: y=22π
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(atan(sqrt(2*x + 3))), divided by x at x->+oo and x ->-oo x→−∞limxatan(2x+3)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the right x→∞limxatan(2x+3)=0 Let's take the limit so, inclined coincides with the horizontal asymptote on the left
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: atan(2x+3)=atan(−2x+3) - No atan(2x+3)=−atan(−2x+3) - No so, the function not is neither even, nor odd