Question

Let S be the set of all functions f : [0, 1] →R which are continuous on [0, 1] and differentiable on (0, 1). Then for every f in S. there exists a c ∈ (0, 1). depending on f, such that

(1) |f(c) - f(1)| < (1 - c)|f'(c)|

(2) |f(c) - f(1)| < |f'(c)|

(3) |f(c) + f(1)| < (1 + c)| f'(c)|

(4) (f(1) - f(c))/(1 - c) = f'(c)

Comments

Please explain more

Answer 1

Option (1), (2), (3) are incorrect for

ƒ(x) = constant and option (4) is incorrect

(f(1) - f(c))/(1 - c) = f'(a) where c < a < 1 (use LMVT)

Also for f(x) = x²  option (4) is incorrect.

Answer 2

The correct option is (3) |f(c) + f(1)| < (1 + c)| f'(c)|

Use LMVT theorem & check option.