find all numbers c that satisfy the conclusion of Rolle's Theorem. (Enter your answers as a comma-separated list.)

Question

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

Answer 1

Thisv function continius on interval [0, 25] and differentiable on interval (0,25). Now, f(0) = f(25) = 0.

So, by rolls theorem exist point 0<, x < 25 where f'(x) = 0. Let find this point x: f'(x) = 1/2x^-1/2 - 1/5 = 0;

1/x^1/2 = 2/5; x^1/2 = 5/2, x = 25/4 = 6.25.

So, f'(6.25) = 0 and 0< 6.25 < 25