Question

Does the function f(x) = 4√x - 3x satisfy the mean value theorem on the interval [4, 49]? If so, find all numbers c on the interval that satisfy the theorem.

Answer 1

The function f(x) = 4√x - 3x satisfies the mean value theorem on the interval [4, 49]. To find the numbers c on the interval that satisfy the theorem, calculate the derivative of f(x), find the average rate of change on the interval, and solve for c.

Step-by-step explanation:

The function f(x) = 4√x - 3x satisfies the mean value theorem on the interval [4, 49].

To find the numbers c on the interval that satisfy the theorem, we first calculate the derivative of f(x). The derivative is given by f'(x) = 2/√x - 3. Next, we find the average rate of change of f(x) on the interval [4, 49] using the formula:

f'(c) = (f(49) - f(4))/(49 - 4)

Finally, we solve for c by setting f'(c) equal to the average rate of change and solve for c.