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Find the minimum of the expression

$E=a^2+2b^2-3a+3b $

$ a,b\in R$

Is there a formula I can apply? How do I find the minumum? Thank you very much in advance!

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    Sorry! English isn't my native language so I didn't exactly know which category this fits in.2012-05-26

2 Answers 2

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You can complete the expressions to a square:

$a^2+2b^2-3a+3b= \left(a-\frac32\right)^2+2\left(b+\frac34\right)^2-\frac94-\frac98$

Since the minimum of a square is 0 and can be attained for $a=\frac32$ and $b=-\frac34$, the minimum of your expression is $-\dfrac{27}8$.

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Assuming you meant $\,a,b,\in\mathbb{R}\,$ , you can use the Hessian matrix of second order derivatives of the

function $\,f(a,b):=a^2+2b^2-3a+3b\,$ to obtain the critical point $\,\displaystyle{\left(\frac{3}{2}\,,\,-\frac{3}{4}\right)}$ , and then check the Hessian at this point is definite positive and thus this point is a minimum one.