Using the first derivative test, find the points of local maxima and local minima of f(x) = x⁴ − 2x², and the local maximum and minimum values.

1. 1 · Write what is given and what is to be found. f(x) = x⁴ − 2x². To find: points of local maxima and minima, and the values there.
2. 2 · Write the formula, rule or theorem. First derivative test: f′ changing from + to − at c gives a local maximum; from − to + gives a local minimum.
3. 3 · Differentiate and find the critical points. f′(x) = 4x³ − 4x = 4x(x − 1)(x + 1) = 0 gives x = −1, 0, 1.
4. 4 · Draw the sign chart. Number line with −1, 0 and 1 marked. Signs of f′ from left to right: −, +, −, +.
5. 5 · Read off the points.
   • x = −1: − to +, local minimum
   • x = 0: + to −, local maximum
   • x = 1: − to +, local minimum
6. 6 · Find the values. f(−1) = 1 − 2 = −1, f(0) = 0, f(1) = 1 − 2 = −1
7. 7 · Check and box the answer. Local maximum value 0 at x = 0. Local minimum value −1 at x = −1 and at x = 1.

Scoring tip: Put each critical point into f(x) for the value. Putting it into f′(x) always gives 0 and loses the mark.