IMathAS-libretexts

Let $\displaystyle {f{{\left({x}\right)}}}=\frac{{1}}{{{2}\sqrt{{{2}\pi}}}}{e}^{{-\frac{{x}^{{2}}}{{8}}}}$ .

Intervals are entered in the format (-00,5)U(7,oo) (these are two infinite intervals). Enter "DNE" if the interval is empty.

Calculate the derivative
$\displaystyle {f}'{\left({x}\right)}=$

Calculate the second derivative
$\displaystyle {f}{''}{\left({x}\right)}=$

List the intervals on which $\displaystyle {f}$ is increasing.

List the intervals on which $\displaystyle {f}$ is decreasing.

List the intervals on which $\displaystyle {f}$ is concave downward.

List the intervals on which $\displaystyle {f}$ is concave upward.

What is the limit as x approaches infinity.
$\displaystyle \lim_{{{x}\to\infty}}{f{{\left({x}\right)}}}=$

What is the limit as x approaches negative infinity.
$\displaystyle \lim_{{{x}\to-\infty}}{f{{\left({x}\right)}}}=$

Sketch the graph and submit with your work.