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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:
f(x) `= x sqrt(1 - x), 0 < x < 1`
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Solution
Given function f(x) `= x sqrt(1 - x), 0 < x < 1` ....(1)
`therefore f'(x) = 1. sqrt(1 - x) + 1/(2 sqrt(1 - x))(- 1) * x`
`= (2 (1 - x) - x)/(2 sqrt(1 - x))`
`= (2 - 3x)/(2 sqrt(1 - x))`
If f'(x) = 0 then `(2 - 3x)/(2 sqrt (1 - x)) = 0,`
`therefore x = 2/3`
At `x = 2/3`, the sign changes from positive to negative when x passes through x `= 2/3`
`therefore` There is a local maximum at the point f
Thus, the local maximum value is f(x) = f `(2/3) = 2/3 sqrt(1 - 2/3) = 2/3 sqrt(1/sqrt3)` ... [Substituting equation (1), x = `2/3,` in (1)]
`= 2/(2 sqrt3) = (2sqrt3)/9`
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