22M.1.AHL.TZ1.8

use of product rule         (M1)

$\frac{dy}{dx}=2\left(4-{\text{e}}^x\right)+2x\left(-{\text{e}}^x\right)$         A1

$=8-2{\text{e}}^x-2x{\text{e}}^x$

[2 marks]

use of product rule         (M1)

$\frac{d^{2}y}{d{x}^2}=-2{\text{e}}^x-2{\text{e}}^x-2x{\text{e}}^x$         A1

$=-4{\text{e}}^x-2x{\text{e}}^x$

$=-2\left(2+x\right){\text{e}}^x$

[2 marks]

$-2\left(2+a\right){\text{e}}^a=0$  OR  sketch of $\frac{d^{2}y}{d{x}^2}$ with $x$-intercept indicated  OR  finding the local maximum of $\frac{dy}{dx}$ at $\left(-2, 8.27\right)$         (M1)

$\left(a=\right) -2$         A1

[2 marks]

Some candidates attempted to apply the product rule in parts (a)(i) and (ii) but often incorrectly, particularly in part (ii) when finding $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$. In part (b) there was little understanding shown of the point of inflexion. There were some attempts, some of which were correct, but many where either the function or the first derivative were set to zero rather than the second derivative.

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