Which number is larger, \[ \pi^3 \qquad\text{or}\qquad 3^\pi? \] Justify your answer.
Solution
1. The function\[ h(x)\coloneqq\frac{\log x}{x} \]is strictly decreasing for \(x>e\).
Differentiation gives \[ h'(x) = \frac{1-\log x}{x^2}. \] If \(x>e\), then \(\log x>1\), so \(h'(x)<0\).
2. One has\[ \frac{\log\pi}{\pi} < \frac{\log3}{3}. \]
The standard inequalities \[ e<3<\pi \] put both \(3\) and \(\pi\) in the interval on which step 1 shows that \(h\) is strictly decreasing. Since \(3<\pi\), \[ h(\pi)<h(3), \] which is the displayed inequality.
3. The larger number is\[ \boxed{3^\pi}. \]
Multiplying the inequality in step 2 by the positive number \(3\pi\) gives \[ 3\log\pi<\pi\log3. \] Exponentiation, which is strictly increasing, yields \[ \pi^3 = e^{3\log\pi} < e^{\pi\log3} = 3^\pi. \]
4. Q.E.D.
Step 3 gives the requested comparison.