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

Question

for the function over the given interval. 

f(x) = 5√x [4, 9]?

Answer 1

The function f(x) = 5√x

Mean value theorem :

If f is

(1) Continuous on closed interval [a, b] where a < b

(2) Differentiable on the open interval (a, b)

then there exist at least one point c in the(a, b) such that f'(c) = [ f(b) - f(a)]/(b - a)

In this case a = 4, b = 9

f(4) = 5(√4) = 10

f(9) = 5(√9) = 15

f'(x) = 5/(2√x)

f'(c)= 5/(2√c)

f'(c) = [15 - 10]/(9 - 4)

= 5/5

= 1

5/(2√c) = 1

25/4c = 1

25 = 4c

c = 25/4

c = 6.25.