Find the local maximum and local minimum values of f(x) = 2x³ − 15x² + 24x + 8.

1. 1 · Write what is given and what is to be found. f(x) = 2x³ − 15x² + 24x + 8. To find: its local maximum and local minimum values.
2. 2 · Write the formula, rule or theorem. Second derivative test: f′(c) = 0 and f″(c) < 0 gives a local maximum; f″(c) > 0 gives a local minimum.
3. 3 · Find the critical points. f′(x) = 6x² − 30x + 24 = 6(x − 1)(x − 4) = 0, so x = 1 or x = 4.
4. 4 · Find f″(x). f″(x) = 12x − 30
5. 5 · Test each point.
   • f″(1) = −18 < 0: local maximum at x = 1
   • f″(4) = 18 > 0: local minimum at x = 4
6. 6 · Find the values.
   • f(1) = 2 − 15 + 24 + 8 = 19
   • f(4) = 128 − 240 + 96 + 8 = −8
7. 7 · Check and box the answer. Local maximum value 19 at x = 1; local minimum value −8 at x = 4.

Scoring tip: Write the sign of f″ at each point in words, e.g. 'f″(1) = −18 < 0, so a local maximum'.