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Thread: 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 $\displaystyle f(x)=ax^2+bx+c$

    $\displaystyle
    f'(x)=2ax+b
    $
    For max or min put $\displaystyle f'(x)=0$

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

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

    ALITER

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