(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]