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Question 13

$$a, b$$ real numbers. The least value of $$a^2 + ab + b^2 - a - 2b$$ is

Solution

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Setting the partial derivatives to zero, $$2a+b-1=0$$ and $$a+2b-2=0$$, gives $$a=0$$, $$b=1$$. Since the quadratic form is positive definite, this is the minimum, and substituting gives $$0+0+1-0-2=-1$$.

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