If `f(x)=sin^6x+cos^6x ,` then range of `f(x)` is `[1/4,1]` (b) `[1/4,3/4]` (c) `[3/4,1]` (d) none of these

Options

A. `[1/4,1]`
B. `[1/4,3/4]`
C. `[3/4,1]`
D. None of these

Answer

A

Explanation

`f(x)=cos^6x+sin^6x`
`=(cos^2x+sin^2x)^3-3cos^2xsin^2x(cos^2x+sin^2x)`
`=1-3cos^2x(1-cos^2x)`
`=3cos^4x-3cos^2x+1`
`=3(cos^4x-cos^2x+1/2)`
`=3((cos^2x=1/2)^2+1/12)`
least value f(x) is `1/4," when "cos^2x-1/2=0`
Greatest value of f(x) is 1, when `cos^2x=0or1`