Satisfy Mean Value Theorom

Question

Verify that the function satisfies the three hypotheses of Rolle's Theorem on the given interval. Then find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)

f(x)=(sqrt of x) - (1/5)(x) , [0,25]

c=

Answer 1

c=25/4

f(x) = sqrx - x/5

f(a) = sqr0 -0/5 = 0

f(b) = sqr25 - 25/5 = 5-5=0

f(a)=f(b) is the extra requirement for Rolle's Theorem

0=0

f'(c) = 1/2sqrc - 1/5 = (f(b)-f(a))/(b-a) = 0/25 = 0

1/2sqrc = 1/5

2sqrc = 5

sqrc = 5/2

c = (5/2)^2 = 25/4

the other two requirements for Rolle's Theorem are the requirements for the Mean Vallue Theorem

that f(x) is continuous on the closed interval [0,25] and differentiable on the open interval (0,25)

Rolles is a special case of the MVT

f'(25/4)= 1/2sqr(25/4) -1/5 = 1/5-1/5 = 0

Answer 2

f(0) = f(25) = 0; so, f'(c) = 1/(2√c) - 1/5 = 0; 2√c = 5; c = 25/4