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The set of all values of a for which the...

The set of all values of a for which the function
`f(x)=((a^2-1)/3)x^3+(a-1)x^2+2x+1`
increases on R, is

A

`(-3 ,1)`

B

`R-[-3,1]`

C

`(-oo,-3)`

D

`[1,oo]`

Text Solution

Verified by Experts

The correct Answer is:
B
If `f(x)=((a^2-1)/(3))x^3a+(a-1)x^2+2x+1` increases on R, then
`f'(x)gt 0 for all x in R`
`rArr (a^2-1)x^2+2(a-1)x+2 gt 0 "for all " x in R`
`rArr a^2 -1 gt 0 and 4(a-1)^1-8(a^2-1)lt 0`
`rArr a^2-1 gt0 and a^2+2a-3 gt 0`
`rArr a in (-oo,-1) cup (1,00) and ain(-oo,-3)cup (1,oo)`
`rArr a in (-oo,-3)cup (1,oo)`
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