IIT-JEE 1981Question 1
Use the function $f(x)=x^{1/x}$, $x>0$, to determine the bigger of the two numbers $e^{\pi}$ and $\pi^{e}$.
Differentiation and its Application
Answer$e^{\pi}$
Solution
Concept. $f(x)=x^{1/x}$ is largest at $x=e$; since $\pi>e$, $\pi^{1/\pi}<e^{1/e}$. Raise both sides to the power $e\pi$. Formulas used. (1) $\dfrac{d}{dx}\,\dfrac{\ln x}{x}=\dfrac{1-\ln x}{x^2}$Step 1 - Monotonic. $\ln f=\frac{\ln x}{x}$. By (1), $f$ increases on $(0,e)$ and decreases on $(e,\infty)$. Step 2 - Compare. $\pi>e$, so $\pi^{1/\pi}<e^{1/e}$. Step 3 - Raise to the power $e\pi$. $\pi^{e}<e^{\pi}$. The bigger number is $e^\pi$. Answer: $e^{\pi}$