Find the maximum and minimum values, if any, of (i) f(x) = 9x² + 6x + 5 and (ii) g(x) = 5 − |x − 2|.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. f(x) = 9x² + 6x + 5 and g(x) = 5 − |x − 2| on R. To find: their maximum and minimum values, if any.
2 · Write the formula, rule or theorem. A square and a modulus are never negative: (expression)² ≥ 0 and |expression| ≥ 0.
3 · Complete the square for f.
- 9x² + 6x + 5 = (9x² + 6x + 1) + 4 = (3x + 1)² + 4
- (3x + 1)² ≥ 0, so f(x) ≥ 4, with f(x) = 4 at x = −1/3.
- f(x) grows without limit, so f has no maximum value.
4 · Use the modulus for g.
- |x − 2| ≥ 0, so g(x) ≤ 5, with g(x) = 5 at x = 2.
- g(x) falls without limit as |x − 2| grows, so g has no minimum value.
5 · Check and box the answer. (i) Minimum value 4 at x = −1/3, no maximum. (ii) Maximum value 5 at x = 2, no minimum.

Scoring tip: Always say where each value occurs, and write 'no maximum' or 'no minimum' in words.