Find the range of `f(x)=(2sin^2x+2sinx+3)/(sin^2x+sinx+1)`

Answer

[7/3, 10/3]

Explanation

`f(x)=(2sin^(2)x+2sinx+3)/(sin^(2)x+sinx+1)`
`=2+(1)/((sinx+(1)/(2))^(2)+(3)/(4))`
`f(x)_(min.)=2+(1)/((9)/(4)+(3)/(4))=(7)/(3)`
`f(x)_(max.)=2+((1)/(3//4))=(10)/(3)`
Hence, range is `[(7)/(3),(10)/(3)]`.