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!
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!
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 $-\dfrac{27}8$.
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.