# How to find the minimum of the expression $E$?

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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Why did you tag your question "regular-expressions"? Also, if you want a calculus-free solution, it would be better to indicate this in the body of your question rather than only in a tag. –  Phira May 26 '12 at 10:05
Sorry! English isn't my native language so I didn't exactly know which category this fits in. –  Grozav Alex Ioan May 26 '12 at 10:59

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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Thanky you! Very helpful! –  Grozav Alex Ioan May 26 '12 at 10:32
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.