Question

Let \(f(x)\) be a derivable function, \(f^{\prime}(x)>f(x)\) and \(f(0)=0\). Then
(A) \(f(x)>0\) for all \(x>0\)
(B) \(f(x)<0\) for all \(x>0\)
(C) no sign of \(f(x)\) can be ascertained
(D) \(f(x)\) is a constant function

Answer 1

Ans : (A)
Hint : Let \(g(x)=e^{-x} f(x) \Rightarrow g^{\prime}(x)=e^{-x}\left(f^{\prime}(x)-f(x)\right)>0\)
\(\Rightarrow g(x)\) is increasing function \(\Rightarrow g(x)>g(0)\) for \(x>0 \Rightarrow e^{-x} f(x)>0\), for \(x>0 \Rightarrow f(x)>0\), for \(x>0\)

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