Question

Let $f(x)=e^{-7x^2}$ .

Then f(x) has a relative minimum at

x=

a relative maximum at

x=

and inflection points at

x=

and at

x=

Write DNE if any of the above do not exist. Write the inflection points (if any) in numerical order, smallest first.

Answer 1

Solution:Domain of given function is $-\infty \:<x<\infty$

$\mathrm{Combine\:the\:critical\:point\left(s\right):}\:x=0\:\mathrm{with\:the\:domain}$

$f(x)=e^{-7x^2}$

$f'(x)=-14xe^{-7x^2}$

$-14xe^{-7x^2}=0$

$x=0$

$\mathrm{The\:function\:monotone\:intervals\:are:}$ $-\infty \:<x<0$ , $0<x<\infty \:$

We should check the sign $f'(x)=-14xe^{-7x^2}$ at each monotone interval

$Summary\:of\:the\:monotone\:intervals\:behavior$ :

1) $-\infty \:<x<0\Rightarrow increasing\Rightarrow+$

2) $0<x<\infty \:$ $\Rightarrow decreasing\Rightarrow -$

3) $x=0 \Rightarrow max \Rightarrow0$

$\mathrm{Plug\:the\:extreme\:point}\:x=0\:\mathrm{into}\:e^{-7x^2}\quad \Rightarrow \quad \:y=1$

f(x) has a relative minimum at:DNErelative maximum at:

$\mathrm{maximum}\left(0,\:1\right)$

inflection points at:

\mathrm{If\:}f\:''\left(x\right)>0\mathrm{\:then\:}f\left(x\right)\mathrm{\:concave\:upwards.}

\mathrm{If\:}f\:''\left(x\right)<0\mathrm{\:then\:}f\left(x\right)\mathrm{\:concave\:downwards.}

$f''(x)=0$

x=$\left(-\frac{\sqrt{14}}{14},\:\frac{1}{\sqrt{e}}\right)$

and at

x=$\left(\frac{\sqrt{14}}{14},\:\frac{1}{\sqrt{e}}\right)$