AP EAMCET · Maths · Differentiation
If \(\mathrm{f}(\mathrm{x})\) is a function such that \(\mathrm{f}^{\prime}(\mathrm{x})=\sqrt{\mathrm{f}^2(\mathrm{x})-1}\) and \(f(0)=1\), then \(f(1)=\)
- A \(\frac{\mathrm{e}^{-2}+1}{2 \mathrm{e}}\)
- B \(\frac{\mathrm{e}^2+1}{2 \mathrm{e}}\)
- C \(\frac{\mathrm{e}^2-1}{2 \mathrm{e}}\)
- D \(\frac{\mathrm{e}^{-2}-1}{2 \mathrm{e}}\)
Answer & Solution
Correct Answer
(B) \(\frac{\mathrm{e}^2+1}{2 \mathrm{e}}\)
Step-by-step Solution
Detailed explanation
\[ \begin{aligned} & \text {} \because \mathrm{f}^{\prime}(\mathrm{x})=\sqrt{\mathrm{f}(\mathrm{x}) \quad 1} \\ & \Rightarrow \frac{\mathrm{f}^{\prime}(\mathrm{x})}{\sqrt{\mathrm{f}^2(\mathrm{x})-1}}=1 \end{aligned} \] Integrating both sides, we get…
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