Question 2(d) Merit

A parametric derivative, a velocity from a log displacement, a second-derivative verification, an inflection point on (ln x)/x, and a turning point that survives an unknown constant.

Consider the function f(x) = (ln x)/x, x > 0. Find the coordinates of the point of inflection on the graph of the function. You can assume that the point you find is a point of inflection.

The 2024 paper, page 7 ↗ Marking schedule, page 3 ↗

Work it out

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

1. 1 A point of inflection comes from which derivative?

   The second: solve f''(x) = 0.

2. 2 First derivative, by the quotient rule.

   f'(x) = (1/x · x − ln x · 1)/x² = (1 − ln x)/x²

3. 3 Now the quotient rule again.

   f''(x) = (−1/x · x² − (1 − ln x) · 2x)/x⁴ = (−3x + 2x ln x)/x⁴ = (2 ln x − 3)/x³

4. 4 Set it to zero and solve.

   2 ln x − 3 = 0, so ln x = 3/2 and x = e^3/2 ≈ 4.4817.

5. 5 The question wants coordinates. Find y.

   y = (ln e^3/2)/e^3/2 = 1.5e^−1.5 ≈ 0.3347

What the marker wanted

• Achieved A correct dy/dx.
• Merit The coordinates of the point of inflection, with both derivatives shown.

In our words, from NZQA’s 2024 marking schedule.

How did it go?

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