5. Graphical Applications of Derivatives

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 $x=1$?

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?

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

Over which interval is the graph of $f\left(x\right)=\frac{1}{2}{x}^2+5x+6$ increasing?

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

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

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

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

For the function $f(x,y)=x^2+y^2-4x-6y+13$, which ordered pair is closest to a local minimum of the function?

Find the exact length of the curve $y=\ln (\sec (x\left)\right)$ for $0\le x\le \frac{\pi }{6}$.

Consider the function $q\left(x\right)=54x^4+125x$. What are the critical numbers of $q\left(x\right)$?

Let $f\left(x\right)=x^3-3x^2+2$. On which of the following intervals is $f\left(x\right)$ decreasing?

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 $v\left(t\right)=t^3-6t^2+9t+2$ 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 $1\le x\le 2$.