Application of Derivatives 2 Question 17
17. 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 $(0 \infty)$
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Answer:
Correct Answer: 17. (a)
Solution:
- $f^{\prime}(x)>2 f(x) \Rightarrow \frac{d y}{y}>2 d x$
$$ \begin{aligned} & \Rightarrow \quad \int _1^{f(x)} \frac{d y}{y}>2 \int _0^{x} d x \\ & \ln (f(x))>2 x \\ & \therefore \quad f(x)>e^{2 x} \end{aligned} $$
Also, as $f^{\prime}(x)>2 f(x)$
$\therefore \quad f^{\prime}(x)>2 c^{2 x}>0$
