Prove that f(x) = x³ + 3x² + 5x − 2 is strictly increasing on R.

1. 1 · Write what is given and what is to be found. Given f(x) = x³ + 3x² + 5x − 2 on R. To prove: f′(x) > 0 for every real x.
2. 2 · Write the formula, rule or theorem. If f′(x) > 0 for every x in an interval, f is strictly increasing on that interval.
3. 3 · Differentiate. f′(x) = 3x² + 6x + 5
4. 4 · Complete the square.
   • 3x² + 6x + 5 = 3(x² + 2x + 1) + 2
   • = 3(x + 1)² + 2
5. 5 · Show that it is positive. (x + 1)² ≥ 0 for every real x, so f′(x) ≥ 2 > 0.
6. 6 · Check and box the answer. f′(x) > 0 for all x in R. Hence f is strictly increasing on R.

Scoring tip: Completing the square is the cleanest proof. If you use the discriminant (36 − 60 < 0), also say that the coefficient of x² is positive.