Least Squares Method 2nd Derivative Negative

Question

I'm having an extremely hard time proving that the 2nd derivative of this equation is negative. Can someone please help me with where I might be wrong?

Answer 1

If for $β^\star$ holds:

$$\frac{d}{dβ}Q(β^\star)=0 \\\frac{d^2}{dβ^2}Q(β^\star)<0,$$

then $β^\star$ can't be a minimum of $Q(β)$. These are the sufficient conditions for a (local) maximum.

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Your computation is correct. Your Professor probably made slight mistake.