Question
Find the local maxima or local minima, if any, of following functions using the first derivative test only. Find also the local maximum and the local minimum values, as the case may be:
f(x) = (x – 1)(x + 2)2
Solution
f (x) = (x – 1)(x + 2)2
= (x – 1), 2 (x + 2) + (x + 2)2, 1 = 2 (x2 + x – 2) + x2 + 4x + 4
= 2 x2 + 2 x – 4 – x2 + 4 x + 4.
= (x – 1), 2 (x + 2) + (x + 2)2, 1 = 2 (x2 + x – 2) + x2 + 4x + 4
= 2 x2 + 2 x – 4 – x2 + 4 x + 4.
f ' (x) = 3 x2 + 6 x
= (x – 1), 2 (x + 2) + (x + 2)2, 1 = 2 (x2 + x – 2) + x2 + 4 x + 4
= 2 x2 + 2 x – 4 – x2 + 4 x + 4.
f ' (x) = 3 x2 + 6 x
f ' (x) = 0 gives us 3 x2 – 6 x = 0. or x2 + 2 x = 0
∴ x (x + 2) = 0 ∴ x = 0, – 2
f ' ' (x) = 6 x + 6
At x = 0, f ' ' (x) = 0 + 6 = 6 > 0
∴ f (x) has a local minimum at x = 0
∴ local minimum value = (c – 1) (0 + 2)2 = – 4
At x = – 2, f ' ' (x) = – 12 + 6 = – 6 < 0
∴ f (x) has a local maximum at x = – 2
∴ local minimum value = (– 2 – 1) (– 2 + 2)2 = 0
