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if a,b,c are real then find the intervia...

if a,b,c are real then find the intervial in which `f(x)=|{:(x+a^2,ab,ac),(ab,x+b^2,bc),(ac,bc,x+c^2):}|` is decreasing.

A

`(-2/3(a^2 + b^2 + c^2),0)`

B

`(0, 2/3 (a^2 + b^2 + c^2))`

C

`(0,(a^2 + b^2 + c^2)/(3))`

D

no where

Text Solution

Verified by Experts

The correct Answer is:
A
`f.(x)=|(1,0,0),(ab,x+b^(2),bc),(ac,bc,x+c^(2)),(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+b^(2),bc),(0,0,1)|`
`=(x+b^(2))(x+b^(2))(x+c^(2))-b^(2)c^(2)+(x+a^(2))(x+c^(2))-a^(2)c^(2)+(x+a^(2))(x+b^(2))-a^(2)b^(2)`
`=3x^(2)+2x(a^(2)+b^(2)+c^(2))lt0`
`f(x)` is decreasing in
`x in((-2)/(3)(a^(2)+b^(2)+c^(2)),0)`
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