Find the intervals in which f(x) = x⁴ − 2x² is strictly increasing or strictly decreasing.

1. 1 · Write what is given and what is to be found. f(x) = x⁴ − 2x². To find: the intervals where f is strictly increasing and where it is strictly decreasing.
2. 2 · Write the formula, rule or theorem. f′(x) > 0 on an interval: strictly increasing. f′(x) < 0: strictly decreasing.
3. 3 · Differentiate and factorise. f′(x) = 4x³ − 4x = 4x(x − 1)(x + 1)
4. 4 · Find the critical points. f′(x) = 0 gives x = −1, 0, 1. They split R into four intervals.
5. 5 · Draw the sign chart. Draw a number line with −1, 0 and 1 marked. Write the sign of f′ above each gap.
6. 6 · Test the sign in each interval.
   • (−∞, −1): negative, so decreasing
   • (−1, 0): positive, so increasing
   • (0, 1): negative, so decreasing
   • (1, ∞): positive, so increasing
7. 7 · Check and box the answer. Strictly increasing on (−1, 0) and (1, ∞). Strictly decreasing on (−∞, −1) and (0, 1).

Scoring tip: Factorise fully before testing signs, and test one point in every interval, not just one.