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]

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?

Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

a. The function f(x) = √x has a local maximum on the interval [0,∞).

if ƒ(x) = 1 / (3x⁴ + 5) , it can be shown that ƒ'(x) = 12x³ / (3x⁴ + 5)² and ƒ"(x) = 180x² (x² + 1) (x + 1) (x - 1) / (3x⁴ + 5)³ . Use these functions to complete the following steps.

d. Identify the local extreme values and inflection points of ƒ .

{Use of Tech} Flow from a tank A cylindrical tank is full at time t=0 when a valve in the bottom of the tank is opened. By Torricelli’s law, the volume of water in the tank after t hours is V=100(200−t)², measured in cubic meters.

d. At what time is the magnitude of the flow rate a minimum? A maximum?  

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.

a. Evaluate g(2), h(2), g'(2), and h'(2).