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

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 $f\left(x\right)=x^3-3x^2+1$. 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?

Let $v\left(t\right)=t^3-6t^2+9t+2$ for $t$ in the interval $[0,5]$. At which value(s) of $t$ does $v\left(t\right)$ attain a local minimum?

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

Let $f(x,y)=x^2+3y^2$. Find the maximum rate of change of $f$ at the point $(1,2)$ and the direction in which it occurs.

Find the exact length of the curve $y=2\surd 3x^{\frac{3}{2}}$ for $0\le x\le 6$.