Let f(x) be a non-negative differentiable function on [0, \infty) such that f(0) = 0 and f'(x) ≤ 2f(x) for all x > 0. Then, on [0, \infty)

Correct Answer is : f(x) is always a constant function

f(x) is always a constant function

KVPY SX SB Exam 2016 Question Paper

Question : 86 of 120

Marks: +1, -0

Let f(x) be a non-negative differentiable function on

[0, ∞)

such that f(0) = 0 and f'(x) ≤ 2f(x) for all x > 0. Then, on

[0, ∞)

Solution:

f'(x) ≤ 2 f(x)

f'(x)e^{–2x} ≤ 2 f(x)e^{–2x}

d/{dx}(f(x)e^{-2n}) ≤ 0

g(x) = f(x)e^{–2x}

is non-Increasing function

x ≥ 0

g(x) ≤ g(0)

f(x)e^{–2x} ≤ f(0)e^{–0}

f(x)e^{–2x} ≤ 0

f(x) ≤ 0

but given f(x) is not negative

∴ f(x) = 0

Constant function

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