Question

How do you prove a differentiable function is strictly increasing if its derivative is nonnegative and not identically equal to zero on any interval?

Answer 1

To prove that a differentiable function is strictly increasing when its derivative is nonnegative and not identically equal to zero on any interval, we can use the Mean Value Theorem.

Step-by-step explanation:

To prove that a differentiable function is strictly increasing when its derivative is nonnegative and not identically equal to zero on any interval, we can use the Mean Value Theorem.

Let f(x) be the differentiable function and f'(x) be its derivative.

Assume that f'(x) is nonnegative and not identically equal to zero on any interval.

By the Mean Value Theorem, if a < b and f(a) < f(b), then there exists c in (a, b) such that f'(c) = (f(b) - f(a))/(b - a) > 0.

Since f'(c) is positive, it implies that f(x) is strictly increasing on the interval (a, b).

Therefore, if the derivative of a differentiable function is nonnegative and not identically equal to zero on any interval, the function is strictly increasing.