Question

The function f(x) is defined for all x and its derivative f ′ (x) has the following properties: 1. f ′ (x) is differentiable for all x, 2. f ′ (1) = f ′ (3) = 0, 3. f ′ (x) has an absolute maximum at x = 2, 4. f ′ (x) decreases on the interval (-[infinity], 2) and increases on the interval (2, [infinity]). Find the following (make sure to justify your answers): (a) The interval(s) where f(x) increases. (b) The interval(s) where f(x) decreases. (c) The local maxima and minima of f(x). (d) The interval(s) where f(x) is concave up. (e) The interval(s) where f(x) is concave down. (f) The points of inflection of f(x).

Answer 1

The function f(x) increases where f'(x) > 0, decreases where f'(x) < 0, and has points of inflection where f''(x) changes sign.

Step-by-step explanation:

Given the properties of the function's derivative f'(x), we can deduce the behavior of the original function f(x). Since f'(1) = f'(3) = 0 and f'(x) decreases on the interval (-∞, 2) and increases on the interval (2, +∞), the function f(x) has local minima at x = 1 and x = 3, and a local maximum at x = 2. Point x = 2 is also a point of inflection, as the behavior of f'(x) changes from decreasing to increasing. The function is concave up when f''(x) > 0, which occurs when f'(x) is increasing (i.e., on (2, +∞)), and concave down when f''(x) < 0, which is when f'(x) is decreasing (i.e., on (-∞, 2)).