If `a ,\ b ,\ c` are real numbers, then find the intervals in which `f(x)=|(x+a^2,a b, a c),( a b, x+b^2,b c),( a c, b c, x+c^2)|` is increasing or decreasing.

Options

A. `(-2/3(a^2+b^2+c^2),0)`
B. `0,(-2/3(a^2+b^2+c^2))`
C. `((a^2+b^2+c^2)/3)`
D. none of these

Answer

A

Explanation

We have
`f(x)=|{:(x+a^2" "ab" "ac),(ab" "x+b^2" "bc),(ac" "bc" "x+c^2):}|`
`f(x)=|{:(1" "0" "0),(ab" "x+b^2" "bc),(ac" "bc" "x+c^2):}|+|{:(x+a^2" "ab" "ac),(" "0" "1" "0),(" "ac" "bc" "x+c^2):}|+ |{:(x+a^2" "ab" "ac),(" "ab" "x+2^2" "bc),(" "0" "0" "1):}|`
`rArr f(x) = 3x^2+2x(a^2+b^2+c^2)`
For f(x) to be decreasing , we must have
f'(x)lt0
`rArr 3x^2+2x(a^2+b^2+c^2)lt0 rArr xsub (-2/3(a^2+b^2+c^2),0)`