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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) = x/2 + 2/x, x > 0`
Solution
Given function g(x) = `x/2 + 2/x, x > 0`
`g'(x) = 1/2 - 2/(x^2), x > 0`
`= (x^2 - 4)/(2x^2)`
`= ((x - 2)(x + 2))/(2x^2)`
g'(x) = 0 ⇒ (x - 2)(x + 2) = 0
∴ x = -2, 2
∴ Critical points are -2 and 2, but x > 0
∴ Critical point = 2
Now, g''(x) = 0 `- (3 xx 2)/x^3 = 6/x^3`
At x = 2, g''(x) = `6/2^3 = 3/4` (positive)
g'' (x) = -2 (-2) x-3 , x > 0 and
g'' (2) = 4(2)-3 = `4/8 = 1/2 > 0`
∴ At x = 2, the value of g(x) is minimum
And minimum value = g(2) = `2/2 + 2/2 = 2`
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