Question

By the mean value theorem, we know there exists at least one ( c ) in the open interval (-6,10) such that ( f'(c) ) is equal to this mean slope. Determine all values of ( c ) that satisfy this condition.

a. ( c = -3 )
b. ( c = 0 )
c. ( c = 5 )
d. ( c = 8 )

Answer 1

The mean value theorem states that there exists at least one value c in the open interval (a, b) such that the derivative of the function, f'(c), is equal to the average rate of change of the function over the interval [a, b]. To find the values of c that satisfy this condition in the given interval (-6, 10), we can evaluate the derivative function and equate it to the mean slope.

Step-by-step explanation:

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

In this case, we are given the open interval (-6, 10) and we need to find the values of c that satisfy the condition f'(c) = mean slope. The mean slope is determined by the formula: (f(b) - f(a))/(b - a), where a = -6 and b = 10.

By evaluating the derivative function f'(x) and equating it to the mean slope, we can find the values of c that satisfy this condition.

Let's go through the steps:

1. Evaluate the function f'(x).
2. Find the mean slope using the formula (f(b) - f(a))/(b - a).
3. Set f'(x) equal to the mean slope and solve for x to find the values of c within the given interval.

By solving this equation, we can determine the values of c that satisfy the given condition.