WBJEE · Maths · Application of Derivatives
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 & Solution
Correct Answer
(A) \(f(x)>0\) for all \(x>0\)
Step-by-step Solution
Detailed explanation
Let \(g(x)=e^{-x} f(x)\) \(\therefore\) \(g^{\prime}(x)=e^{-x} f^{\prime}(x)-e^{-x} f(x)\) \(\quad=e^{-x}\left(f^{\prime}(x)-f(x)\right)\) Since, \(f^{\prime}(x)>f(x),\) so \(f^{\prime}(x)-f(x)>0\) \(\therefore e^{-x}\left(f^{\prime}(x)-f(x)\right)>0\)…
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