I know you have to find the 1st and 2nd derivative. But I'm not completely sure what to do after that. How do you find the critical numbers and where to plot the points?
$\displaystyle y = xe^x$
first observations ...
obviously passes thru the origin
$\displaystyle y \to 0$ as $\displaystyle x \to -\infty$
$\displaystyle y$ increases w/o bound as $\displaystyle x \to \infty$
$\displaystyle y' = e^x(x+1)$
critical value at $\displaystyle x = -1 $
for $\displaystyle x < -1$ , $\displaystyle y' < 0$ ... $\displaystyle y$ is decreasing
for $\displaystyle x > -1$ , $\displaystyle y' > 0$ ... $\displaystyle y$ is increasing
absolute minimum at $\displaystyle \left(-1, -\frac{1}{e}\right)$
$\displaystyle y'' = e^x(x+2)$
critical value at $\displaystyle x = -2$
for $\displaystyle x < -2$ , $\displaystyle y'' < 0$ ... $\displaystyle y$ is concave down
for $\displaystyle x > -2$ , $\displaystyle y'' > 0$ ... $\displaystyle y$ is concave up
inflection point at $\displaystyle \left(-2 , -\frac{2}{e^2}\right)$
graph should look as attached ...