Find the ratio of greatest value of `2-cos x+ sin^(2)x` to its least value.

Options

A. `7/4`
B. `11/4`
C. `13/4`
D. none of these

Answer

C

Explanation

Let `y=2-cosx+sin^(2)x.Then,`
` y=2-cosx+1-cos^(2)x`
`impliesy=3-(cos^(2)x+cos)impliesy=13/4-(cosx+1/2)^(2)`
Now, `-1lecosx le1` for all x
`implies-1/2le(cosx+1/2)^(2)lt0"for all x"`
`implies13/4-9/4le13/4(cosx+1/2)^(2)le13/4"for all x"`
`implies1ley13/4`
`thereforey_(max)=13/4and y_(min)=1`
Hence, required ratio is `13/4`