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//2`
B. `1//2`
C. `1//4`
D. `-3//2`

Answer

D

Explanation

f(x)`=sin^(6)x+cos^(6)x+k (sin^4x+cos(4)x)`
f(x)` =sin^6x+cos^6x+k(sin^4x+cos^4 x)`
`=(sin ^2x)^3+cos^2 x)^3+k (sin^2 x)+(cos^2 )^2]`
`=(sin ^2x)^3+(cos ^2 x)^3-3 sin^2x.cos^2x(sin^2 x+ cos^2)`
`+ k[ sin^x +cos^2 x)^2-2sin^2 x. cos^2x]`
`=(1-3 sin^2 x cos^2x)+ k[1-2 sin^2 x cos^2 x]`
f(x) is constant if k = -3/2.