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The value of the function (x-1)(x-2)^2 a...

The value of the function `(x-1)(x-2)^2` at its maxima is

A

1

B

2

C

0

D

`(4)/(27)`

Text Solution

Verified by Experts

The correct Answer is:
D
Given `f(x)=(x-1)(x-2)^(2)`
`f(x)=(x-1)(x^(2)+4-4x),f(x)=(x^(3)-5x^(2)+8x-4)`
Now `f^(`)(x)=3x^(2)-10x+8,f^(`)(x)=0`
`implies3x^(2)-10x+8=0implies(3x-4)(x-2)=0implies(4)/(3),2`
Now `f^(``)(x)=6x-10`
`f^(``)((4)/(3))=6xx(4)/(3)-10lt0`
`f^(``)(2)=12-10gt0`
Hence at `x=(4)/(3)` the function will occupy maximum value maximum value `=f((4)/(3)=(4)/(27)`
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