Question

Let f(x) = 2sinx mean value theorem |f'(x)| <= ?

Answer 1

The mean value theorem states that if a function is continuous and differentiable, there exists at least one point where the derivative is equal to the average rate of change. For f(x) = 2sin(x), f'(x) = 2cos(x). The maximum value of |cos(x)| is 1, so |f'(x)| ≤ 2.

Step-by-step explanation:

The mean value theorem states that if a function is continuous on a closed interval [a, b] and differentiable on the open interval (a, b), then there exists at least one point c in (a, b) where the derivative of the function is equal to the average rate of change of the function over the interval [a, b].

In this case, f(x) = 2sin(x), so f'(x) = 2cos(x). To find the maximum value of |f'(x)|, we need to find the maximum value of |cos(x)|. The maximum value of |cos(x)| is 1, so we have |f'(x)| ≤ 2.