Application Of Derivatives Ques 34

34. If $f: R \rightarrow R$ is a differentiable function such that $f^{\prime}(x)>2 f(x)$ for all $x \in R$, and $f(0)=1$ then

(2017 Adv.)

(a) $f(x)>e^{2 x}$ in $(0, \infty)$

(b) $f^{\prime}(x)<e^{2 x}$ in $(0, \infty)$

(c) $f(x)$ is increasing in $(0, \infty)$

(d) $f(x)$ is decreasing in $(0, \infty)$

Show Answer

Answer:

Correct Answer: 34.(c, d)

Solution:

Formula:

Increasing and decreasing of a function:

$f^{\prime}(x)>2 f(x) \Rightarrow \frac{dy}{y}>2 dx$

$ \begin{aligned} & \Rightarrow \quad \int_{1}^{f(x)} \frac{d y}{y}>2 \int_{0}^{x} d x \\ & \ln (f(x))>2 x \\ & \therefore \quad f(x)>e^{2 x} \end{aligned} $

Also, as $f^{\prime}(x)>2 f(x)$

$\therefore \quad f^{\prime}(x)>2 c^{2 x}>0$

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