155. Which is bigger, πᵉ or e^π?
Calculators have taken some of the mystery out of this once-challenging question.
(Go ahead and check; you will see that it is a very close call.)
You can answer the question without a calculator, though.
d. Conclude that
xᵉ < eˣfor all positivex ≠ e.

Verified step by step guidance

1. Rewrite the expressions \( \pi^{e} \) and \( e^{\pi} \) in a form that allows comparison using logarithms. Consider taking the natural logarithm of both expressions to compare their sizes without directly calculating their values.
2. Express the comparison as \( \pi^{e} < e^{\pi} \) if and only if \( e \ln(\pi) < \pi \ln(e) \). Since \( \ln(e) = 1 \), this simplifies to comparing \( e \ln(\pi) \) and \( \pi \).
3. Define a function \( f(x) = \frac{\ln(x)}{x} \) for \( x > 0 \) to analyze the inequality \( x^{e} < e^{x} \) for \( x \neq e \). The inequality \( x^{e} < e^{x} \) is equivalent to \( e \ln(x) < x \), or \( \frac{\ln(x)}{x} < \frac{1}{e} \).
4. Find the critical points of \( f(x) = \frac{\ln(x)}{x} \) by differentiating: \( f'(x) = \frac{1 - \ln(x)}{x^{2}} \). Set \( f'(x) = 0 \) to find that the maximum occurs at \( x = e \).
5. Conclude that since \( f(x) \) attains its maximum at \( x = e \), for all positive \( x \neq e \), \( f(x) < f(e) = \frac{1}{e} \). Therefore, \( x^{e} < e^{x} \) for all positive \( x \neq e \), which includes the comparison between \( \pi^{e} \) and \( e^{\pi} \).

Key Concepts

Exponential and Power Functions

Understanding the difference between expressions like πᵉ and e^π requires familiarity with power functions (x raised to a constant) and exponential functions (constant raised to variable). Recognizing how these functions behave for positive real numbers is essential to compare their values without direct computation.

Function Comparison Using Logarithms

Comparing expressions like πᵉ and e^π can be simplified by taking natural logarithms, converting powers into products. This technique transforms the inequality into a comparison of products involving logarithms, making it easier to analyze without a calculator.

Monotonicity and Critical Points of the Function f(x) = x^{1/x}

The function f(x) = x^{1/x} reaches its maximum at x = e, which helps prove inequalities like xᵉ < eˣ for x ≠ e. Understanding how to find and interpret critical points and monotonicity of such functions is key to concluding the given inequality.