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:
Solution
Let f (x) = – (x – 1)3 ( x + 1 )2
∴ f ' (x) = – [(x – 1)3. 2(x + 1) + (x + 1)2. 3(x – 1)2]
= – (x – 1)2 (x + 1) [2 (x – 1) + 3 (x + 1)] = – (x – 1)2 (x + 1) (5 x + 1)
f ' (x) = 0 ⇒ – (x – 1)2 (x + 1) (5 x + 1) = 0 
.
When x < 1 slightly, f ' (x) = – [( + )( + )( + )] = – ve
When x > 1 slightly, f ' (x) = – [( + )( + )( + )] = – ve
∴ at x = 1 , f '(x) does not change sign ,
∴ x = 1 is a point of inflexion.
When x < – 1 slightly. f ' (x) = – [( + )( – )( – )] = – ve
When x > – 1 slightly. f ' (x) = – [( + )( + ) ( – )] = + ve
∴ at x = – 1, f ' (x) changes from – ve to + ve
∴ f (x) has local minimum value at x = – 1 and this local minimum value = 0.
