Question
Find all points of local maxima and local minima of the function f given by f (x) = x3 – 3x + 3.
Solution
f (x) = x3 – 3 x + 3
∴ f ' (x) – 3 x2 3 = 3 (x – 1) ( v + 1)
f ' (x) = 0 ⇒ 3 x2 – 3 = 0 ⇒ x2 – 1 – 0 ⇒ x2 = 1 ⇒ v = – 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
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
