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 $x=c$?

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 $x=1$?

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?

Let $f\left(x\right)=x^3-3x^2+1$. What is the absolute minimum value of $f$ on the closed interval $[0,3]$?

Given the function $f\left(x\right)=x^3-12x$, on which interval is the function increasing?

Suppose $f\left(x\right)$ is continuous on the interval $[0,4]$ and $f\left(x\right)>0$ for all $x$ in $(1,3)$. Which of the following could be the entire interval over which $f\left(x\right)$ is positive?

Find the exact length of the curve $y=\frac{x^3}{3}+\frac{1}{4x}$ for $1\le x\le 2$.