Question

Let the function f be continuous and differentiable for all x. Suppose you are given that f(−1)=−3, and that f'(x) for all values of x. Use the Mean Value Theorem to determine the largest possible value of f(5).

Answer 1

To determine the largest possible value of f(5) using the Mean Value Theorem, we need to find a point c in the interval [-1,5] such that the derivative of f at c is equal to the average rate of change of f over the interval.

Step-by-step explanation:

To determine the largest possible value of f(5) using the Mean Value Theorem, we first need to find a point c in the interval [-1,5] such that the derivative of f at c is equal to the average rate of change of f over the interval. Since f is continuous and differentiable for all x, we can apply the Mean Value Theorem.

Let's call the point c the critical point, and let's call the average rate of change of f over the interval the slope of the secant line. According to the Mean Value Theorem, there exists a point c in the interval [-1,5] such that f'(c) = (f(5) - f(-1))/(5 - (-1)).

Since f(-1) = -3 and f'(x) for all values of x, we can substitute these values into the equation: f'(c) = (f(5) - (-3))/(5 - (-1)). Solving for f(5), we get f(5) = f'(c) * (5 - (-1)) + (-3).