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

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

Quantrex Academy · Accepted Answer

Correct answer: A. f ( x ) is always a constant function

f ^ ( x ) 2 f ( x ) f ^ ( x ) e ^ -2 x 2 f ( x ) e ^ -2 x d dx ( f ( x ) e ^ -2 n ) 0 ~g ( x )= f ( x ) e ^ -2 x is non-Increasing function x 0 ~g ( x ) g (0) f ( x ) e ^ -2 x f (0) e ⁻⁰ f ( x ) e ^ -2 x 0 f ( x ) 0 but given f ( x ) is not negative f ( x )=0 Constant function

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

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