Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval.

Question

Find the number c that satisfies the conclusion of the Mean Value Theorem on the given interval. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.)

f(x) =x , [0, 25] c =

Answer 1

f(0) = 0 and f(25)=5

derivative f'(x) = (1/2)x^(-1/2)

the slope of the secant line is (5-0)/(25-0) = 5/25 = 1/5

MVT says there exists c in [0,25] such that f'(c) = 1/5

(1/2) c^(-1/2) = 1/5

5 c^(-1/2) = 2

c^(-1/2) = 2/5

1 = (2/5) c^(1/2)

5/2 = c^(1/2)

25/4 = c