Find the local maxima and local minima, if any, of the following functions. Find also the local maximum and the local minimum values, as the case may be:

g(x) = x3 – 3x

g(x) = x3 – 3x

⇒ g’(x) = 3x2 - 3


Now, g’(x) = 0


⇒ 3x2 - 3 = 0


⇒ 3x2 = 3


⇒ x = � 1


g’’(x) = 6x


Now, g’(1) = 6>0


and g’(-1) = -6 < 0


Then, by second derivative test,


⇒ x = 1 is point of local maxima and local minima of g at x = 1 is


g(1) = 13 – 3 = 1-3 =-2


And,


x = -1 is point of local maxima and local maximum value of g at x = -1 is


g(-1) = (-1)3 – 3(-1) = -1+3 = 2


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