5. Graphical Applications of Derivatives

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:

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?

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

Determine where the local and absolute maxima and minima occur on the given graph of $f\left(x\right)$.

{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.

b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.  

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 3x² - 4x + 2

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x³ / 3 - 9x

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = 3x³ + 3x² / 2 - 2x

Locating critical points Find the critical points of the following functions. Assume a is a nonzero constant.

ƒ(x) = x³ -4a²x