Question 2(c) Merit

A quotient with cos, a cot gradient, a stationary point on an exponential quotient, inflection on x² ln x, and a related-rates chase where the rate given is an angle.

A curve is defined by f(x) = e^x/(x² + 2x). Find the x-value(s) of any point(s) on the curve where the tangent is parallel to the x-axis.

The 2023 paper, page 7 ↗ Marking schedule, page 4 ↗

Work it out

Try it on paper first. Stuck? Take one hint at a time.

1. 1 A tangent parallel to the x-axis has what gradient?

   Zero, so solve f'(x) = 0.

2. 2 Differentiate with the quotient rule.

   f'(x) = (e^x(x² + 2x) − e^x(2x + 2))/(x² + 2x)² = (e^x(x² − 2))/(x² + 2x)²

3. 3 Set it to zero.

   e^x is never zero, so x² − 2 = 0 and x = ±√2 ≈ ±1.41. Both are allowed: the curve is undefined only at x = 0 and x = −2.

What the marker wanted

• Achieved The correct derivative.
• Merit Both values of x, from the derivative.

In our words, from NZQA’s 2023 marking schedule.

How did it go?

Got it moves it further out. Review brings it back tomorrow.