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Math Help - The quadratic function

  1. #1
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    Exclamation 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?

    Thanks in advance.
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  2. #2
    Senior Member pankaj's Avatar
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    Let f(x)=ax^2+bx+c

     <br />
f'(x)=2ax+b<br />
    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} <br />
{2a} is point of minima and if 2a<0 then x=- <br />
\frac{b}{2a} is point of maxima

    ALITER

     <br />
f(x)=ax^2+bx+c=a(x+\frac{b}{2a})^2+\frac{4ac-b^2}{4a}<br />
    Obviously, f(x)\geq\frac{4ac-b^2}{4a} if a>0
    f(x)\leq\frac{4ac-b^2}{4a} if a<0
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