Range of `f(theta)=cos^2theta(cos^2theta+1)+2sin^2theta` is `[3/4,1]` (b) `[3/(16),1]` (c) `[3/4,7/4]` (d) `[7/4,2]`

Options

A. `[3//4,1]`
B. `[3//16,1]`
C. `[3//4,7//4]`
D. `[7//4,2]`

Answer

D

Explanation

`f(theta)=cos^2theta(cos^2theta+1)+2sin^2theta`
`=cos^4theta+cos^2theta+sin^2theta+sin^2theta`
`=cos^4theta+1+1-cos^2theta`
`=(cos^2theta-1/2)^2+2-1/4`
`=(cos^2theta-1/2)^2+7/4`
`f_("min")=7/4`
`f_("max")=1/4+7/4=2`