# Math Help - The quadratic function

Well, we know that a quadratic function is a MIN when the coefficient of x^2 is positive and a MAX when the coefficient of x^2 is negative. Is there a particular proof or derivation for this? What is the reasoning behind it?

2. Let $f(x)=ax^2+bx+c$

$
f'(x)=2ax+b
$

For max or min put $f'(x)=0$

$2ax+b=0$ ..i.e $x=-\frac{b}{2a}$

$f''(-\frac{b}{2a})=2a$. If $2a>0$ then $x=-\frac{b}
{2a}$
is point of minima and if $2a<0$ then $x=-
\frac{b}{2a}$
is point of maxima

ALITER

$
f(x)=ax^2+bx+c=a(x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a}
$

Obviously, $f(x)\geq\frac{4ac-b^2}{4a}$ if $a>0$
$f(x)\leq\frac{4ac-b^2}{4a}$ if $a<0$