A function \(f\) is defined by \(f(x) = {x^3} + {{\text{e}}^x} + 1,{\text{ }}x \in \mathbb{R}\). By considering \(f'(x)\) determine whether \(f\) is a one-to-one or a many-to-one function.

\(f'(x) = 3{x^2} + {{\text{e}}^x}\)     A1

Note:     Accept labelled diagram showing the graph \(y = f'(x)\) above the x-axis;

do not accept unlabelled graphs nor graph of \(y = f(x)\).

EITHER

this is always \( > 0\)     R1

so the function is (strictly) increasing     R1

and thus \(1 - 1\)     A1

OR

this is always \( > 0\;\;\;{\text{(accept }} \ne 0{\text{)}}\)     R1

so there are no turning points     R1

and thus \(1 - 1\)     A1

Note:     A1 is dependent on the first R1.

[4 marks]

The differentiation was normally completed correctly, but then a large number did not realise what was required to determine the type of the original function. Most candidates scored 1/4 and wrote explanations that showed little or no understanding of the relation between first derivative and the given function. For example, it was common to see comments about horizontal and vertical line tests but applied to the incorrect function.In term of mathematical language, it was noted that candidates used many terms incorrectly showing no knowledge of the meaning of terms like ‘parabola’, ‘even’ or ‘odd’ ( or no idea about these concepts).