Question

If the function $f(x) = \left(\frac{1}{x}\right)^{2x}; \, x>0$ attains the maximum value at $x = \frac{1}{c}$, then:

• A. $e^\pi<\pi^c$
• B. $e^{2\pi}<(2\pi)^c$
• C. $e^\pi>\pi^c$
• D. $(2e)^\pi>\pi^{(2e)}$

Answer 1

The Correct Option is C

Step 1 : Initialize \(y = \left( \frac{1}{x} \right)^{2x}\). Apply the natural logarithm to both sides, resulting in \(\ln y = 2x \ln \left( \frac{1}{x} \right)\). Simplifying yields \(\ln y = -2x \ln x\).
Step 2: Differentiate with respect to \(x\). Differentiating both sides with respect to \(x\) yields \(\frac{1}{y} \frac{dy}{dx} = -2(1 + \ln x)\). Multiply by \(y\) to obtain \(\frac{dy}{dx} = y \cdot (-2)(1 + \ln x)\).
Step 3: Analyze function behavior. The function \(f^n\) is decreasing for \(x > \frac{1}{e}\). This establishes the following inequalities: \(e < \pi\), \(\left( \frac{1}{e} \right)^{2e} > \left( \frac{1}{\pi} \right)^{2\pi}\), and \(e^\pi > \pi^e\).