Understanding the Maximum of the Function f(x) = x^1/x

The problem states that the function \(f(x) = x^{1/x}\) for \(x > 0\) has its maximum value at \(x = e\). This is a crucial piece of information for Comparing e^pi and pi^e.

The fact that \(f(x)\) has a maximum value at \(x = e\) means that for any other value of \(x > 0\) where \(x \ne e\), the value of \(f(x)\) must be strictly less than the value of \(f(e)\). Mathematically, this can be written as:

\(f(x) < f(e)\) for all \(x > 0, x \ne e\).

Substituting the function definition, we get:

\(x^{1/x} < e^{1/e}\) for all \(x > 0, x \ne e\).

Applying the Maximum Property to Compare e^π and π^e

We need to compare \(e^\pi\) and \(\pi^e\). These expressions involve the numbers \(e\) and \(\pi\). We know that \(e \approx 2.718\) and \(\pi \approx 3.141\). Both \(e\) and \(\pi\) are positive numbers, and importantly, \(\pi \ne e\).

Since \(\pi > 0\) and \(\pi \ne e\), we can use the property of the function \(f(x) = x^{1/x}\) that we discussed. We know that \(f(\pi) < f(e)\).

Let's write this using the function definition:

\(\pi^{1/\pi} < e^{1/e}\)

Now, our goal is to relate this inequality to \(e^\pi\) and \(\pi^e\). We can raise both sides of the inequality to the power of \(e\pi\). Since \(e > 0\) and \(\pi > 0\), \(e\pi\) is a positive number, so raising to this power will preserve the direction of the inequality.

\((\pi^{1/\pi})^{e\pi} < (e^{1/e})^{e\pi}\)

Using the power rule for exponents \((a^b)^c = a^{bc}\), we simplify both sides:

\(\pi^{(1/\pi) \cdot (e\pi)} < e^{(1/e) \cdot (e\pi)}\)

\(\pi^e < e^\pi\)

Conclusion: Comparing e^π and π^e

The inequality we derived is \(\pi^e < e^\pi\). This is equivalent to \(e^\pi > \pi^e\).

This shows how knowing the point of maximum value for the function \(f(x) = x^{1/x}\) allows us to solve problems involving the comparison of expressions like \(e^\pi\) and \(\pi^e\). This type of mathematical analysis is common in calculus.

Let's check the options provided:

• Option 1: \(e^\pi > \pi^e\)
• Option 2: \(e^\pi < \pi^e\)
• Option 3: \(e^\pi = \pi^e\)
• Option 4: \(e^\pi \le \pi^e\)

Our derived inequality \(e^\pi > \pi^e\) matches Option 1. This confirms the result obtained by analyzing the maximum value of the function \(x^{1/x}\).