October 2020 Paper 3 Q11

11

The questions in this section refer to the article on the Insert. You should read the article before attempting the questions.

The relevant parts of the article “Which is bigger?” are reproduced below; the line numbers are those printed on the Insert.

Line 38
\(\ln\pi < \dfrac{\pi}{\mathrm{e}}\)

Line 39
\(\mathrm{e}^x\) is an increasing function for all values of \(x\)

Line 40
hence \(\pi < \mathrm{e}^{\frac{\pi}{\mathrm{e}}}\)

Lines 41–42
Assuming that the usual rules of indices apply to irrational powers of irrational numbers, raising both sides of the inequality to the power e gives the desired result.

Show that \(\mathrm{e}^x\) is an increasing function for all values of \(x\), as stated in line 39. [2]