Question:
If $f: R \rightarrow R$ is a differentiable function such that $f'(x)>2 f(x)$ for all $x \in R$, and $f(0)=1$, then
If $f: R \rightarrow R$ is a differentiable function such that $f'(x)>2 f(x)$ for all $x \in R$, and $f(0)=1$, then
Updated On: June 02, 2025
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The Correct Option is C
Solution and Explanation
$f'(x)-2 f(x)>0$
$\Rightarrow \frac{d}{d x}\left(f(x) \cdot e^{-2 x}\right)>0$
$\Rightarrow f(x) \cdot e^{-2 x}$ is increasing function
$\Rightarrow f(x) \cdot e^{-2 x}>f(0)=1 \,\,\,\,\forall x>0$
$f(x)>e^{2 x} \forall x>0$
$\Rightarrow f'(x)>2 f(x)>2 e^{2 x}>0 \forall x>0$
$\Rightarrow \frac{d}{d x}\left(f(x) \cdot e^{-2 x}\right)>0$
$\Rightarrow f(x) \cdot e^{-2 x}$ is increasing function
$\Rightarrow f(x) \cdot e^{-2 x}>f(0)=1 \,\,\,\,\forall x>0$
$f(x)>e^{2 x} \forall x>0$
$\Rightarrow f'(x)>2 f(x)>2 e^{2 x}>0 \forall x>0$
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