5. Graphical Applications of Derivatives

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

d. Give the approximate coordinates of the zero(s) of f.

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

a. Find the critical points of f and determine where f is increasing and where it is decreasing.

Use the graphs of ƒ' and ƒ" to complete the following steps. <IMAGE>

c. Determine where f has local maxima and minima.

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

f(x) = √(9 - x²) + sin⁻¹ (x/3)

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

f(x) = x ln x - 2x + 3 on (0,∞)

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

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

f(x) = x² - 2 ln x

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

f(x) = -2x⁴ + x² + 10

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

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

f(x) = xe⁻(ˣ²/₂)

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

f(x) = tan⁻¹ (x/(x²+2))

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?

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,∞)

Suppose the position of an object moving horizontally after seconds is given by the function s(t) = 32t - t⁴, where 0 ≤ t ≤ 3 and s is measured in feet, with s > 0 corresponding to positions to the right of the origin. When is the object farthest to the right?

Absolute maxima and minima Determine the location and value of the absolute extreme values of ƒ on the given interval, if they exist.

ƒ(x) = 2x³ - 15x² + 24x on [0,5]