Let f : [a, b] → R be such that f is differentiable in (a, b), f i

1.	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)
Answer» 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)

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)

Answer»

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)

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