Question

Suppose that ƒ has a positive derivative for all values of x and that ƒ(1) = 0. Which of the following statements must be true of the function g(x) =|x0 f(t) dt?

Give reasons for your answers.
a. g is a differentiable function of x.
b. g is a continuous function of x.
c. The graph of g has a horizontal tangent at x = 1.
d. g has a local maximum at x = 1.
e. g has a local minimum at x = 1.
f. The graph of g has an inflection point at x = 1.
g. The graph of dg>dx crosses the x-axis at x = 1.

Answer 1

The function g(x) is differentiable and continuous, has a horizontal tangent and a local minimum at x = 1. Statements b, c, and e are true, while statements a is contextually true, d and f are false, and g is true regarding g(x).

Step-by-step explanation:

Regarding the function g(x) = ∫1x f(t) dt, if f has a positive derivative for all values of x and that f(1) = 0, we can derive certain truths about the function g(x).

• g is both differentiable and continuous, because the integral of a continuous function results in a differentiable and continuous function (g is a continuous function).
• Since f has a positive derivative, f is increasing, which means g(x) will increase as x increases from 1. Because f(1) = 0, the graph of g will have a horizontal tangent at x = 1.
• g will have a local minimum at x = 1 since f(1) = 0 and f is increasing for all x. This makes x = 1 the lowest point on the graph of g around that area.

Therefore, statements b, c, and e must be true. Statement a is true because the integral of a differentiable function is also differentiable, stating d is false because x = 1 represents a minimum, not a maximum, statement f is false because there is no information provided about a change in concavity at x = 1, and statement g is true as dg/dx at x = 1 is the same as f(1), which is zero, indicating the graph of dg/dx will cross the x-axis at x = 1.