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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))`.

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AI Generated Solution

To prove that \( f'(x) = 3x^2 + 2x(a^2 + b^2 + c^2) \) for the function given by the determinant: \[ f(x) = \left| \begin{array}{ccc} x + a^2 & ab & ac \\ ab & x + b^2 & bc \\ ac & bc & x + c^2 \end{array} \right| ...
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