Let $f\left(x\right)=x^3+a{x}^2$. For what value of $a$ does $f\left(x\right)$ have a critical point at $x=1$?

Suppose the graph of $f\left(x\right)$ is shown below. At which of the following intervals is $f^{″}\left(x\right)$ $>0$?

Which of the following best describes the end behavior of the function $f\left(x\right)=2x^3-3x^2-3$?

For which values of $x$ is the function $f\left(x\right)=(x^2-3)e^{-x}$ increasing?

Consider the following graph of $f\left(x\right)$. Which of the following points are inflection points of $f$?

For the curve $y=5\ln \left(x\right)$, at what point does the curve have maximum curvature?

What is the end behavior of the graph of the polynomial function $y=10x^9-4x$?

Let $f\left(x\right)$ = $x^3-3x^2+4$ on the interval $[0,3]$. What is the maximum value of $f$ on this interval?

Suppose a continuous function $f\left(x\right)$ is defined on the closed interval $[a,b]$, and $x=c$ is a critical point in $(a,b)$. Which of the following is true about the curve at the point $x=c$?