Find the approximate value(s) of c guaranteed by the Mean Value Theorem

Question

Find the approximate value(s) of c guaranteed by the Mean Value Theorem for Integrals for the function  over the domain interval .

Answer 1

For f(x) = 4x - x² on [0, 4], then f(0) = 0 and f(4) = 0 so we can find the slope of the secant line using [f(4) - f(0)]/(4 - 0) which is 0

For f(x) = 4x - x² then f '(x) = 4 - 2x and f '(x) = 0 when x = 2 therefore the value of c = 2 is the value guaranteed by the mean value theorem.