2025HSCShort answer

Q16 (a)

4 marks

Question

Consider the function

f(x)=\frac{x^2}{e^x}.

Find the stationary points of the function and determine their nature.

Try the question before revealing the answer.

Solution

Criteria	Marks
Provides correct solution	4
Finds both stationary points	3
Finds one of the stationary points, or equivalent merit	2
Attempts to use quotient rule, or equivalent merit	1

Sample answer:

\begin{aligned} f(x)&=\frac{x^2}{e^x}\\ f'(x)&=\frac{2xe^x-x^2e^x}{e^{2x}}=\frac{xe^x(2-x)}{e^{2x}}\\ f'(x)&=\frac{x(2-x)}{e^x}\\ x&=0,\quad 2-x=0\\ x&=0,\quad x=2\\ y&=0,\quad y=\frac{4}{e^2}=0.541\ldots \end{aligned}

$x$	$-1$	$0$	$1$	$2$	$3$
$f'(x)$	Negative	0	Positive	0	Negative

Minimum at $(0,0)$ and maximum at $\left(2,\frac{4}{e^2}\right)$ .

Supplied marking guidance · human reviewed

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Syllabus alignment

Calculus

Differential CalculusApply the 2017 Mathematics Advanced syllabus content for differential calculus.

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