Let f : [a, b] → R be such that f is differentiable in (a, b), f is continuous at x = a and x = b and moreover f(a) = 0 = f(b). Then

(A) there exists at least one point c in (a, b) such that f'(c) = f(c)

(B) f'(x) = f(x) does not hold at any point in (a, b)

(C) at every point of (a, b), f'(x) > f(x)

(D) at every point of (a, b), f'(x) < f(x)

The correct option (A) there exists at least one point c in (a, b) such that f'(c) = f(c)Explanation:Let h(x) = e

^{–x}f(x) h(a) = 0,h(b) = 0 h(x) is continuous and diff. by rolles theorem

h'(c) = 0, c ∈ (a, b)

e

^{–x}f(x) + (–e^{–x})f(x) = 0e

^{–c}f'(c) = e^{–c}f(c)f'(c) = f(c)