TS EAMCET · Maths · Basic of Mathematics

\(\mathrm{f}: \mathbb{R} \rightarrow \mathbb{R}\) is a function defined by \(f(x)=\frac{1}{e^x+2 e^{-x}}\) Assertion (A) : \(\mathrm{f}(\mathrm{c})=\frac{1}{3}\) for some values of \(\mathrm{c} \in \mathrm{R}\) Reason (R): \(0 < \mathrm{f}(\mathrm{x}) \leq \frac{1}{2 \sqrt{2}}\) for all \(\mathrm{x} \in \mathrm{R}\) Then which of the following options is correct?

A (A) and (R) are true. (R) is the correct explanation of (A)
B (A) and (R) are true, but (R) is not the correct explanation for \((\mathrm{A})\)
C (A) is true but \((R)\) is false
D (A) is false but (R) is true

Verified Solution

Answer & Solution

Correct Answer

(A) (A) and (R) are true. (R) is the correct explanation of (A)

Step-by-step Solution

Detailed explanation

\begin{aligned} & \quad f(x)=\frac{1}{e^x+2 e^{-x}} \\ & \because \quad \frac{e^x+2 e^{-x}}{2} \geq \sqrt{e^x \times 2 e^{-x}} \\ & e^x+2 e^{-x} \geq 2 \sqrt{2} \\ & \therefore \quad \frac{1}{e^x+2 e^{-x}} \leq 2 \sqrt{2} \\ & \because \quad f(x)>0 \\ & \therefore \quad 0 < f(x)…

Apply the relevant identity from the chapter and substitute the values established in the previous step, keeping the original constraint in view.

Use the simplified expression to isolate the unknown, then plug the result back into the question setup to confirm the working is internally consistent.

Cross-check against a boundary or limiting case. Both the dimensional balance and the qualitative behaviour at the extreme should agree with the candidate answer.

Combine the steps above to identify the correct option. The remaining choices each fail at least one of the consistency, dimensional, or boundary checks.

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