Calculus I, Fall 2021

Brief solutions to Quiz 5

Nov 23, 2021

1. (30 pts) Find absolute minimum and absolute maximum of f(x) = \frac{1}{x} + ln x on the interval [0.5, 4]. For your reference, ln 2 ≈ 0.6931. Ans:

f′(x) = \frac{−1}{x^{2}} + \frac{1}{x} = \frac{x − 1}{x^{2}}.

Critical points in (0.5, 4): x = 1 only. Compare values of f(0.5), f(1) and f(4) shows that f(1) = absolute minimum, f(4) = absolute maximum.

2. (20+20 pts) Suppose that f′′ is continuous on [a, b] and f has 3 distinct zeros on [a, b]. Show that f′′ has at least one zero in (a, b). First state clearly the content of theorem you use (need not prove the theorem). Ans: Rolle’s Theorem or Mean Value Theorem (both applicable): see textbook. Apply Roll’s Theorem to f to get two distinct zeros of f′. Then apply Roll’s Theorem again to f′ to get a zero of f′′.

3. (30 pts) Find all critical points of f(x) = x^{\frac{2}{3}} (x − 4). For each one of them, use first derivative test to determine whether it corresponds to a local minimum, a local maximum or neither. Ans: See page 9 of Lecture 14 note.