Let `f(x) = sin^6x + cos^6x + k(sin^4 x + cos^4 x)` for some real number k. Determine(a) all real numbers k for which `f(x)` is constant for all values of x.

Options

A. [-1,0]
B. `[0,1/2]`
C. `[-1,-1/2]`
D. None of these

Answer

C

Explanation

f(x)=0
`rArr (1-3 ) sin^2 x cos^2 x)+k[1-2 sin^2 x cos^2 x]=0`
`rArr K+1 =(sin ^2 x cos^2 x)/(1-2 sin^2x cos^2x)`
`rArrk=(3sin^2xcos^2x-1)/(1-2sin^2xcos^2x)`
`=-3/2(1-2sin^2xcos^2x-1/3)/(1-2sin^2xcos^2x)`
`=-3/2(1-(1/3)/(1-2sin^2xcos^2x))`
minimum of `sin^2xcos^2x " is 0 at " x=0,pi//2`
Maximum of `sin^2xcos^2x" is "1//4" at " x=pi//2`
Hence, k in[1,-1/2]