19M.2.SL.TZ1.S_3

pestleMathematicsAISLPaper 219M· sl-4-1-concepts-reliability-and-sampling-techniquessource ↗

Consider the function $f\left( x \right) = {x^2}{{\text{e}}^{3x}}$ , $x \in \mathbb{R}$ .

Find $f'\left( x \right)$ .

[4]

The graph of $f$ has a horizontal tangent line at $x = 0$ and at $x = a$ . Find $a$ .

[2]

Markscheme / solution

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

choosing product rule (M1)

eg $uv' + vu'$ , ${\left( {{x^2}} \right)^\prime }\left( {{{\text{e}}^{3x}}} \right) + {\left( {{{\text{e}}^{3x}}} \right)^\prime }{x^2}$

correct derivatives (must be seen in the rule) A1A1

eg $2x$ , ${\text{3}}{{\text{e}}^{3x}}$

$f'\left( x \right) = 2x{{\text{e}}^{3x}} + 3{x^2}{{\text{e}}^{3x}}$ A1 N4

[4 marks]

valid method (M1)

eg $f'\left( x \right) = 0$ , ,

$a = - 0.667\left( { = - \frac{2}{3}} \right)$ (accept $x = - 0.667$ ) A1 N2

[2 marks]

Examiners’ report

[N/A]