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Question
Find the absolute maximum value and the absolute minimum value of the following function in the given interval:
`f(x) = 4x - 1/x x^2, x in [-2 ,9/2]`
Solution
Given function f(x) = 4x `- 1/2 x^2,` interval `[-2,9/2]`
∴ f'(x) = 4 - `1/2`. 2x = 4 - x
If f'(x) = 0, then 4 - x = 0 ⇒ x = 4
At x = -2, f(-2) = 4 (-2) - `1/2 (-2)^2`
`= - 8 - 1/2 xx 4`
= - 8 - 2
= - 10
At x = 4, `f(4) = 4(4) - (4)^2/2`
`= 16 - 16/2`
= 16 - 8
= 8
At x = `9/2`, `f (9/2) = 4 xx 9/2 - 1/2 xx 81/4`
`= 18 - 81/8`
`= (144 - 81)/8`
`= 63/8`
= 7.875
∴ Absolute maximum value of f (x) = 8 at x = 4
Absolute minimum value of f (x) = -10 at x = -2
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