3. 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:
 (ii) g(x) = x ^3 - 3x

Answers (1)

Given function is 
g(x) = x ^3 - 3x\\ g^{'}(x) = 3x^2 - 3\\ g^{'}(x)=0\Rightarrow 3x^2-3 =0 \Rightarrow x = \pm 1\\
Hence, the critical points are 1 and - 1
Now, by second derivative test 
g^{''}(x)=6x
g^{''}(1)=6 > 0
Hence, 1 is the point of minima and the minimum value is 
g(1) = (1)^3 - 3(1) = 1 - 3 = -2
g^{''}(-1)=-6 < 0
Hence, -1 is the point of maxima and the maximum value is 
g(1) = (-1)^3 - 3(-1) = -1 + 3 = 2

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