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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) = 1/(x^2 + 2)`
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Solution
Given, `g (x) = 1/ (x^2 + 2)`
= `g' (x) = (-2x)/(x^2 + 2)^2`
For critical points, let g' (x) = 0
= `(-2x)/(x^2 + 2)^2 = 0`
x = 0 ....(∵ x2 + 2 ≠ 0)
`g'' (x) = (6x^2 - 4)/(x^2 + 2)^3; g'' (0) = (-4)/8 <0`
∴ g has a local maximum at x = 0 and local
maximum value is `g (0) = 1/ (0 + 2) = 1/2`
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