Methods2026 NHT Exam 1 · Q9Tech-free9 marks

Stationary point, sign and integral of an exponential quotient

Let $f : \R \to \R,\ f(x) = \dfrac{2 - e^{x}}{e^{2x}}$ .

Show that $f'(x) = \dfrac{e^{x} - 4}{e^{2x}}$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

b.i.

The graph of $y = f(x)$ has exactly one stationary point.

Find the coordinates of this stationary point.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

b.ii.

Determine the nature of this stationary point.

2 marks

Hints, answer checking and worked solutions come with a Season Pass.

Find all values of $x$ such that $f(x) \geq 0$ .

1 mark

Hints, answer checking and worked solutions come with a Season Pass.

Given that $\int_p^q f(x)dx = 0$ , where $p \neq q$ and $p,\ q \in \R$ , find an expression for $p$ in terms of $q$ .

3 marks

Hints, answer checking and worked solutions come with a Season Pass.