Question

If the local maximum value of the function $f(x) = \left(\frac{\sqrt{3e}}{2\sin x}\right)^{\sin^2 x}, x \in \left(0, \frac{\pi}{2}\right) \text{ is } \frac{k}{e},$ then the value of k is

• A. e^{7/8}
• B. e^{11/8}
• C. e^{9/8}
• D. e^{13/8}

Answer 1

Answer: (B)

e^{11/8}

Answer 2

Let $y = f(x)$. Taking the natural logarithm, we get $\ln y = \sin^2 x \ln\left(\frac{\sqrt{3e}}{2\sin x}\right)$. Simplify $\ln y = \sin^2 x \left(\ln(\sqrt{3e}) - \ln(2\sin x)\right) = \sin^2 x \left(\frac{1}{2}\ln(3e) - \ln 2 - \ln(\sin x)\right)$. Differentiate with respect to $x$: $\frac{f'(x)}{f(x)} = \sin(2x) \left(\frac{1}{2}\ln 3 + \frac{1}{2} - \ln 2 - \ln(\sin x)\right) - \sin x \cos x$. Setting $f'(x)=0$ (and since $f(x) > 0$), we set the term in parentheses to zero after dividing by $\sin x \cos x$: $2\left(\frac{1}{2}\ln 3 + \frac{1}{2} - \ln 2 - \ln(\sin x)\right) - 1 = 0$, which simplifies to $\ln(\sin^2 x) = \ln(3/4)$. Thus, $\sin^2 x = 3/4$. Since $x \in (0, \pi/2)$, $\sin x = \sqrt{3}/2$, which means $x = \pi/3$. The local maximum value is $f(\pi/3) = \left(\frac{\sqrt{3e}}{2(\sqrt{3}/2)}\right)^{3/4} = (\sqrt{e})^{3/4} = e^{3/8}$. Given this value is $k/e$, we have $e^{3/8} = k/e$, so $k = e \cdot e^{3/8} = e^{11/8}$.