Find a quadratic polynomial `varphi(x)` whose zeros are the maximum and minimum values of the function `f(x)=|(1+sin^2x, cos^2x,sin2x) ,(sin^2x,1+cos^2x,sin2x), (sin^2x ,cos^2x,1+sin2x)|`

Options

A. `alpha+ beta^9=4`
B. `alpha ^3 - beta ^7 =26`
C. `alpha^(2n)-beta ^(2n)` is always and even integer for `n in N`
D. a triangle can be costructed having its sides as `alpha , beta and alpha- beta`

Answer

D

Explanation

Applying `C_ 1 rarr C+ C_2` we get
`f(x)=|{:(1+sin^2" " x cos^2x " " sin2x),(sin^2x " "1+cos^2 x " "sin 2x),(sin^2x " "cos^2x" "1+sin 2x):}|`
`=|(2 " "cos^x " " sin 2x),(0" "1 " "0),(-1 " "0" "1)| " " " Applying " R_2 rarr R_2-R_1 and R_3 rArr R_3-R_1`
`=2+ sin 2x `
We know that
`-1 le sin 2x le 1 " for all " x`
`rArr 1 le 2 + sin 2x le 3 " for all " x `
`rArr 1 le f(x) le 3 "for all " x`
`rArr alpha =3 and beta =1`
Clearly , these values of `alpha and beta` satisfy (a),(b) and (c)