The critical points of the function
`f(x)=2sin ^(2)((x)/(6))+sin((x)/(3))-((x)/(3))`
whose coordinates satisfy the inequality `x^(2)-10lt-19.5x` is

Options

A. `-6pi`
B. `6pi`
C. `(9pi)/(2)`
D. `-4pi`

Answer

A

Explanation

`f'(x)=2((1)/(3))sin""((x)/(6))cos""((x)/(6))`
`+((1)/(3))cos ""((x)/(3))-((1)/(2))`
`=1/3[2sin""((x)/(6))cos ""((x)/(6))-sin^(2)""((x)/(6))]`
Put `f'(x)=harrsin""((x)/(6))=0`
`or tan""((x)/(6))=1`
`harrx/6=npi+(pi)/(4)nninI`
`x^(2)-10lt-19.5x`
`harr(x+9.75)^(2)lt105.0625`
`harr (x-0.5)(x+20)lt0`
`implies-20lt x lt0.5`
So, the cirtical points satisfying the last inequation will be `0, -6pi-(9pi)/(2).`