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
- $
- $
- $>
- $
The Correct Option is C
Solution and Explanation
$\Rightarrow \frac{d}{d x}\left(f(x) \cdot e^{-2 x}\right)>0$
$\Rightarrow f(x) \cdot e^{-2 x}$ 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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Various Applications of Derivatives-
Rate of Change of Quantities:
If some other quantity ‘y’ causes some change in a quantity of surely ‘x’, in view of the fact that an equation of the form y = f(x) gets consistently pleased, i.e, ‘y’ is a function of ‘x’ then the rate of change of ‘y’ related to ‘x’ is to be given by
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Increasing and Decreasing Function:
Consider y = f(x) be a differentiable function (whose derivative exists at all points in the domain) in an interval x = (a,b).
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- Likewise, if for any two points x1 and x2 in the interval x such a manner that x1 < x2, there holds an inequality f(x1) ≥ f(x2); then the function f(x) is known as decreasing in this interval.
- The functions are commonly known as strictly increasing or decreasing functions, given the inequalities are strict: f(x1) < f(x2) for strictly increasing and f(x1) > f(x2) for strictly decreasing.
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