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Calculus: Graphical, Numerical, Algebraic
Which is Bigger, $\pi^{e}$ or $e^{\pi} ?$ Calculators have taken some of the mystery out of this once-challenging question. (Go ahead and check; you will see that it is a surprisingly close call.) You can answer the question without a calculator, though, by using he result from Example 3 of this section.
Recall from that example that the line through the origin tangent to the graph of $y=\ln x$ has slope 1$/ e$ .
(a) Find an equation for this tangent line.
(b) Give an argument based on the graphs of $y=\ln x$ and the tangent line to explain why $\ln x<x / e$ for all positive $x \neq e$
(c) Show that $\ln \left(x^{2}\right)<x$ for all positive $x \neq e$ .
(d) Conclude that $x^{e}<e^{x}$ for all positive $x \neq e$ .
(e) So which is bigger, $\pi^{e}$ or $e^{\pi} ?$