Find a quadratic polynomial `phi(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]|`

Explanation

Given,

`f(x)=|[1+sin^2x,cos^2x,sin2x],[sin^2x,1+cos^2x,sin2x],[sin^2x,cos^2x,1+sin2x]|`

Applying,`R_1->R_1​−R_2,​R_2​->R_2​−R_3`

​`f(x)=|[1,-1,0],[0,1,-1],[sin^2x,cos^2x,1+sin2x]|`

=`1|[1,-1],[cos^2x,1+sin2x]|+1|[0,-1],[sin^2x,1+sin2x]|`

=`1+4sin2x+cos^2x+sin^2x`

=`2+4sin2x`

f(x) will be maximum when sin2x is maximum and we know that the maximum value of sin2x is 1

...