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:
Let f (x) = x3 + 3 x.
∴ f ' (x) = 3x2 – 3 = 3 (x – 1) (x + 1)
f ' (x) = 0 ⇒ 3 x2 – 3 = 0
∴ x2 – 1 = 0 ⇒ x2 = 1 ⇒ x = – 1, 1
When x < – 1 slightly, f ' (x) = 3 (– ve) (– ve ) = + ve
When x > – 1 slightly, f ' (x) = 3 (– ve) (+ ve) = – ve
∴ at x = – 1 , f ' (x) changes from + ve to – ve
∴ f (x) has local maximum value at x = – 1
and this local maximum value = (– 1 )3 – 3 (– 1) = – 1 + 3 = 2
When x < 1 slightly, f ' (x) = 3 (– ve) ( + ve) = – ve
When x > 1 slightly, f ' (x) = 3 ( + ve) (+ ve) = + 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 = (1)3 – 3 (1) = 1 – 3 = – 2
