5. Graphical Applications of Derivatives

Which of the following best explains why the function $f\left(x\right)=\frac{1}{x-2}$ is discontinuous at $a=2$?

The position function of a particle is given by $s\left(t\right)=t^3-6t^2+9t+2$. At what time $t$ is the speed of the particle minimum?

Which of the following could be a turning point for the continuous function $f\left(x\right)$?

Let the function be defined by $f\left(x\right)=x^3-3x^2+2$. At what value(s) of $x$ does $f\left(x\right)$ have a relative maximum?

For the curve $y=7\ln \left(x\right)$, at what value of $x$ does the curve have maximum curvature?

Let $f\left(x\right)$ = $\left\{\begin{array}{cc}ax+b& \text{,}x<2\\ x^2& \text{,}x\ge 2\end{array}\right\}$. For which values of $a$ and $b$ is $f\left(x\right)$ continuous everywhere?

Consider the graph of $f^{\prime }\left(x\right)$ below. How many local maxima does $f\left(x\right)$ have?

Which of the following is a possible turning point for the continuous function $f\left(x\right)$?