5. Graphical Applications of Derivatives

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

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

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

ƒ(t) = t/ t² + 1

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

ƒ(x) = eˣ + e⁻ˣ

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

ƒ(x) = 1 / x + ln x

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

ƒ(x) = x² √(x + 5)

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

ƒ(x) = x √(x-a)

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

ƒ(t) = 1/5 t⁵ - a⁴t

Sketch the graph of a continuous function ƒ on [0, 4] satisfying the given properties.

ƒ' (x) = 0 for x = 1 and 2; ƒ has an absolute maximum at x = 4; ƒ has an absolute minimum at x= 0; and ƒ has a local minimum at x = 2.