| Date | None Specimen | Marks available | 3 | Reference code | SPNone.1.hl.TZ0.9 |
| Level | HL only | Paper | 1 | Time zone | TZ0 |
| Command term | Sketch | Question number | 9 | Adapted from | N/A |
Question
The function f is defined on the domain \(x \geqslant 0\) by \(f(x) = {{\text{e}}^x} - {x^{\text{e}}}\) .
(i) Find an expression for \(f'(x)\) .
(ii) Given that the equation \(f'(x) = 0\) has two roots, state their values.
Sketch the graph of f , showing clearly the coordinates of the maximum and minimum.
Hence show that \({{\text{e}}^\pi } > {\pi ^{\text{e}}}\) .
Markscheme
(i) \(f'(x) = {{\text{e}}^x} - {\text{e}}{x^{{\text{e}} - 1}}\) A1
(ii) by inspection the two roots are 1, e A1A1
[3 marks]
A3
Note: Award A1 for maximum, A1 for minimum and A1 for general shape.
[3 marks]
from the graph: \({{\text{e}}^x} > {x^{\text{e}}}\) for all \(x > 0\) except x = e R1
putting \(x = \pi \) , conclude that \({{\text{e}}^\pi } > {\pi ^{\text{e}}}\) AG
[1 mark]
