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 $f\left(x\right)$?

Which of the following statements is true about the absolute maximum and minimum values of a continuous function $f\left(x\right)$ on a closed interval $[a,b]$?

In the context of extrema, if all the rates of change $f^{\prime }\left(x\right)$ (derivatives) in a set of problems are negative, what does this indicate about the behavior of the functions involved?

Which of the following best describes the difference between a relative maximum and an absolute maximum of a function on an interval $I$?

Given the function $f\left(t\right)=t^3-3t^2+2$, for which values of $t$ is the curve concave upward? (Select the correct interval.)

For the function $f\left(x\right)=-x^2+4x+1$, at which $x$-value does a local maximum occur?

Finding Extreme Values

In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.

y = 𝓍³ ― 2𝓍 + 4

A tangent line approximation of a function value is an underestimate when the function is: