Answer to Question #155402 in Calculus for berkant

Question

Sketch the graph of

y=1/π+r2

by finding the domain, symmetries, critical points, inflection points,

intercept points, asymptotes, extremas, intervals on which the function is increasing or decreasing,

concave up or down.

Answer 1

$y=\frac{1}{\pi+r^2}$

Domain: $r\isin(-\infin,\infin)$

Asymptote:

$\displaystyle\lim_{r\to -\infin}f(r)=\displaystyle\lim_{r\to \infin}f(r)=0$

Horizontal asymptote is $y=0$

Symmetry:

there is symmetry respect to y-axis: $f(r)=f(-r)$

Critical point, extrema:

$y'=-\frac{2r}{(\pi+r^2)^2}=0\implies r=0$

the function is increasing on $r\isin(-\infin,0)$ , $y'>0$

the function is decreasing on $r\isin(0,\infin)$ , $y'<0$

maxima is $(0,1/\pi)$

Inflection points:

$y''=-\frac{2(\pi+r^2)^2-2r\cdot4r(\pi+r^2)}{(\pi+r^2)^4}=0$

$4r^2-\pi-r^2=0$

$r=\pm\sqrt{\pi/3}$

Inflection points are $(\sqrt{\pi/3},\frac{1}{\pi+3}), (-\sqrt{\pi/3},\frac{1}{\pi+3})$

Intercept point:

$r=0\implies y=1/\pi$

y-intercept point is $(0,1/\pi)$

there no x-intercepts