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Question
Find the local maxima and local minima, if any, of the following function. Find also the local maximum and the local minimum values, as the case may be:
g(x) = x3 − 3x
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Solution
Given function g(x) = x3 - 3x
`therefore g'(x) = 3x^2 - 3`
if, g'(x) = 0 and 3x2 - 3 = 0
⇒ x2 - 1 = 0
⇒ x = `pm` 1
The points at which extremum may occurs are -1 and +1.
g' (x) = 6x
g' (-1) = 6 (-1) = -6 < 0
∴ g has a local maximum at x = -1 and local maxum value at x = -1 is g (-1) = (-1)3 - 3 (-1)
= -1 + 3
= 2
g' (1) = 6 × 1
= 6 > 0
∴g has a local minimum at x = 1 and local minimum value at x = 1 is g (1)
= 13 - 3 × 1
= -2
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