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"If "f(x)=|{:(x+a^(2),ab,ac),(ab, x+b^(2...

`"If "f(x)=|{:(x+a^(2),ab,ac),(ab, x+b^(2),bc),(ac,bc, x+c^(2)):}|," then prove that "`
`f'(x)=3x^(2)+2x(a^(2)+b^(2)+c^(2))`.

Text Solution

Verified by Experts

We have
`f(x)=|{:(x+a^(2),ab,ac),(ab,x+b^(2),bc),(ac,bc,x+c^(2)):}|`
`therefore" "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+b^(2),bc),(0,0,1):}|`
`=|{:(x+b^(2),bc),(bc,x+c^(2)):}|+|{:(x+a^(2),ac),(ac,x+c^(2)):}|+|{:(x+a^(2),ab),(ab,x+b^(2)):}|`
`=[(x+b^(2))(x+c^(2))-b^(2)c^(2)]+[(x+a^(2))(x+c^(2))-a^(2)c^(2)]`
`=3x^(2)+2x(a^(2)+b^(2)+c^(2))`
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