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The function f: R-R is defined by f(x)=c...

The function `f: R-R` is defined by `f(x)=cos^2x+sin^4xforx in Rdot` Then the range of `f(x)` is `(3/(4,1)]` (b) `[3/(4,1))` (c) `[3/(4,1)]` (d) `(3/(4,1))`

A

`((3)/(4),1]`

B

`[(3)/(4),1)`

C

`[(3)/(4),1]`

D

`((3)/(4),1)`

Text Solution

Verified by Experts

The correct Answer is:
C
`y=f(x)=cos^(2)x+sin^(4)x`
`=cos^(2)x+sin^(2)x(1-cos^(2)x)`
`=cos^(2)x+sin^(2)x-sin^(2)x cos^(2)x`
`=1-sin^(2)x cos^(2)x`
`=1-(1)/(4) sin^(2)2x`
` :. (3)/(4) le f(x) le 1 " " ( :' 0 le sin^(2)2x le 1)`
` :. f(x) in [3//4,1]`
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