(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 ${{\text{e}}^\pi } > {\pi ^{\text{e}}}$ .

(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]