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–xf(x) h(a) = 0,
h(b) = 0 h(x) is continuous and diff. by rolles theorem
h'(c) = 0, c ∈ (a, b)
e–xf(x) + (–e–x)f(x) = 0
e–cf'(c) = e–cf(c)
f'(c) = f(c)