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

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

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?

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?

Given the function $f(x,y)=x^2-y^2$, which of the following statements correctly describes its local maxima, local minima, and saddle points?

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 $[a,b]$?

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?

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