5. Graphical Applications of Derivatives

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.

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

ƒ' (x) and ƒ'3 are undefined; ƒ'(2) = 0; has a local maximum at x= 1; ƒ has local minimum at x = 2; and ƒ has an absolute maximum at x= 3; and ƒ has an absolute minimum at x = 4 .

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

Use the following graphs to identify the points on the interval [a, b] at which local and absolute extreme values occur. <IMAGE>

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.

f(x) = (x - 1)²