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Minimum value of f(x)=cos^(2)x+(secx)/(4...

Minimum value of `f(x)=cos^(2)x+(secx)/(4)`, `x in (-(pi)/(2),(pi)/(2))` is

A

`3//2`

B

`3//4`

C

`3//8`

D

none of these

Text Solution

Verified by Experts

The correct Answer is:
B
`(b)` `f(x)=cos^(2)x+(secx)/(4)`
`=cos^(2)x+(1)/(4cosx)`
For `x in (-(pi)/(2),(pi)/(2))`, `cosx gt 0`
Now `f(x)=cos^(2)x+(1)/(8cosx)+(1)/(8cosx)`
Using `A.M. ge G.M.`
`implies(cos^(2)x+(1)/(8cosx)+(1)/(8cosx))/(3) ge ((1)/(8^(2)))^((1)/(3))=[((1)/(4))^(3)]^((1)/(3)`
`impliescos^(2)x+(1)/(4cosx) ge (3)/(4)`
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