5. Graphical Applications of Derivatives

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

c. Give the approximate coordinates of the inflection point(s) of f.

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

f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8

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>

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

ƒ(x) = eˣ + e⁻ˣ

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = sin x − cos x,0 ≤ x ≤ 2π

38. What values of a and b make f(x) = x^3 + ax^2 + bx have

b. a local minimum at x = 4 and a point of inflection at x = 1?

{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.  

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>

Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.

f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)

Trajectory high point A stone is launched vertically upward from a cliff 192 ft above the ground at a speed of 64 ft/s. Its height above the ground t seconds after the launch is given by s = -16t² + 64t + 192, for 0 ≤ t ≤ 6. When does the stone reach its maximum height?

Identifying Extrema

In Exercises 41–52:

a. Identify the function’s local extreme values in the given domain, and say where they occur.

f(x) = (x + 1)², −∞ < x ≤ 0

Identifying Extrema

In Exercises 15–18:

a. Find the open intervals on which the function is increasing and those on which it is decreasing.

b. Identify the function’s local and absolute extreme values, if any, saying where they occur.

Identifying Extrema

In Exercises 53–60:

a. Find the local extrema of each function on the given interval, and say where they occur.

f(x) = csc²x − 2cot x, 0 < x < π

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

f(x) = -12x⁵ + 75x⁴ - 80x³

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.