Chapter5 In-Class Quiz

The graph of a function f is shown beside. Which of the following could be the graph of the derivative function f'?

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The polynomial function f has a second derivative

function f′′. Several values of f′′ are given in the table.

Which statement must be true?

• f changes concavity in the interval (2, 4).

• f has a point of inflection at x = 3.

• f has a local maximum at x = 3.

• f is not increasing on the interval (0, 2).

The function f is continuous for all real numbers and has a relative maximum at (−4, 3) and a relative minimum at (1, −8). Which of the following statements must be true?

• f has a point of inflection between (−4, 3) and (−1, 8).

• f has a horizontal asymptote at either (−4, 3) or (1, −8).

• f has at least one zero.

• f is decreasing on the open interval (−4, 1).

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Suppose f is a polynomial function of degree greater than 2 and f(a) = f(b) where a < b. Then which of the following statements must be true for at least one number c in the interval (a, b)?

I. f(c) = 0

II. f'(c) = 0

III. f''(x) = 0

• I only

• II only

• II and III only

• I, II, and III

Find the absolute maximum value of the function

f(θ) = cos θ − cos²θ on the closed interval [0, π].

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A water storage tank in the shape of a right circular cylinder is constructed so that the sum of the height and the circumference of the tank is 300 m. What are the radius r and the height h of the tank with maximum volume?

• radius: 50 m;

  height 200 m

• radius: 100π m;

  height 100 m

• No critical point

• Point of inflection

• Local minimum

• Local maximum