SATHEE: Application of Derivatives 2 Question 17

Application of Derivatives 2 Question 17

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

(2017 Adv.)

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

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

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

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Answer:

Correct Answer: 17. (a)

Solution:

$f^{'} (x) > 2 f (x) \Rightarrow \frac{d y}{y} > 2 d x$

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

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

$∴ f^{'} (x) > 2 c^{2 x} > 0$

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Application of Derivatives 2 Question 17

17. If f:R→R is a differentiable function such that f′(x)>2f(x) for all x∈R, and f(0)=1 then

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