Question

Find all numbers c [4,12] for which the derivative is equal to the slope of the secant line connecting the function endpoints. If there is more than one such number c, use a comma to separate your answers. Round to the nearest thousandth. If there is no such number c, type "does not exist" in the entry box.

Answer 1

To find the numbers c [4,12] for which the derivative is equal to the slope of the secant line connecting the function endpoints, find the derivative of the function and set it equal to the slope of the secant line. List the values of c that satisfy this condition, separated by a comma. If no such number exists, state that it does not exist.

Step-by-step explanation:

To find all the numbers c [4,12] for which the derivative is equal to the slope of the secant line connecting the function endpoints, we need to find the derivative of the function. Let's say the function is f(x). Then, we need to find the derivative of f(x) and set it equal to the slope of the secant line between the points (4, f(4)) and (12, f(12)). If there is more than one value of c that satisfies this condition, we will list them separated by a comma. If no such number exists, we will state that it does not exist.

Answer 2

Step-by-step explanation:

`f'(x) =(1)/(2√(x-3))`

By MVT, there exist a number c on [4, 12] such that

`f'(c) = (f(12)-f(4))/(12-4)`

`f(12) = 3, and, f(4) = 1, hence f'(c) = (1)/(4)`

therefore,

`(1)/(2√(x-3)) = (1)/(4)`

Solve this equation, you have x =1.

However, this is out of the given interval. Hence does not exist