Browse by Stream
QnA
Home
QnA
Go To Your Study Dashboard

Get Answers to all your Questions

header-bg qa
  • Home
  • Engineering
  • Let  be a differentiable function such that f(1)=e and 

Let f :(0, \infty )\rightarrow (0,\infty ) be a differentiable function such that f(1)=e and \lim_{t\rightarrow x}\frac{t^{2}f^{2}(x)-x^{2}f^{2}(t)}{t-x}=0. If f(x)=1, then x is equal to:
Option: 1 \frac{1}{e}
Option: 2 2e
Option: 3 \frac{1}{2e}
Option: 4 e

Answers (1)

best_answer

\begin{aligned} &L=\operatorname{Lim}_{t \rightarrow x} \frac{t^{2} f^{2}(x)-x^{2} f^{2}(t)}{t-x}\\ &\text { using L H'opital. rule }\\ &\mathrm{L}=\operatorname{Lim}_{\mathrm{t} \rightarrow \mathrm{x}} \frac{2 \mathrm{tf}^{2}(\mathrm{x})-\mathrm{x}^{2} \cdot 2 \mathrm{f}^{\prime}(\mathrm{t}) \cdot \mathrm{f}(\mathrm{t})}{1}\\ &\Rightarrow \mathrm{L}=2 \mathrm{xf}(\mathrm{x})(\mathrm{f}(\mathrm{x})-\mathrm{xf}(\mathrm{x}))=0(\text { given }) \end{aligned}

\\\Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{xf}^{\prime}(\mathrm{x}) \Rightarrow \int \frac{\mathrm{f}^{\prime}(\mathrm{x}) \mathrm{dx}}{\mathrm{f}(\mathrm{x})}=\int \frac{\mathrm{dx}}{\mathrm{x}} \\ \Rightarrow \ell \mathrm{n} \operatorname{lf}(\mathrm{x})|=\ell \mathrm{n}| \mathrm{x} \mid+\mathrm{C} \\ \because \mathrm{f}(1)=\mathrm{e}, \mathrm{x}>0, \mathrm{f}(\mathrm{x})>0 \\ \Rightarrow \mathrm{f}(\mathrm{x})=\mathrm{ex}, \text { if } \mathrm{f}(\mathrm{x})=1 \Rightarrow \mathrm{x}=\frac{1}{\mathrm{e}}

Posted by

himanshu.meshram

View full answer

JEE Main high-scoring chapters and topics

Study 40% syllabus and score up to 100% marks in JEE

Similar Questions

Trending Articles/News

anam-khan

JEE Main Result 2024 Live: JEE April results at jeemain.nta.ac.in likely today; cutoff, rank predictor, update
anam-khan

JEE Main 2024 session 2 final answer key out at jeemain.nta.ac.in; steps to check probable scores
anam-khan

JEE Main Result 2024 Live: Download session 2 final answer key at jeemain.nta.ac.in; rank predictor, cutoff

Latest Question