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The maximum value of the function f(x)=3...

The maximum value of the function `f(x)=3x^3-18x^2+27x-40" on the set " S={x in R: x^2+30 le 11x}` is

A

122

B

`-122`

C

`-222`

D

222

Text Solution

Verified by Experts

The correct Answer is:
A
We have,
`f(x)=3x^(3)-18x^(2)+27x-40`
`rArr f'(x)=9x^(2)-36x+27`
`=9(x^(2)-4x+3)=9(x-1)(x-3)" " ....(i)`
Also, we have `S={x in R:x^(2)+30 le11x}`
Clearly, `x^(2)+30 le 11x`
`rArr x^(2)-11+30 le0`
`rArr (x-5)(x-6)le 0 rArr x in [5,6]`
So, S=[5,6]
Not that f(x) inscreasing in [5,6]
`[ :. f'(x) le 0 "for" x in [5,6]`
`:. f((6)` is maximum, where
`f(6)=3(6)^(3)-18(6)^(2)+27-40=122`
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