Use calculus techniques (can they be a bit more specific? ) to show that the graph of the quadratic function f(x) = ax2 + bx + c, a>0 is decreasing on the interval x<(-b/2a) and increasing on the interval x>(-b/2a).

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- Apr 20th 2008, 01:33 PM #1

- Apr 20th 2008, 02:34 PM #2
well lets start with what we know and some calculus.

$\displaystyle f(x)=ax^2+bx+c$ let take the derivative

$\displaystyle f'(x)=2ax+b$ set the derivative equal to zero.

$\displaystyle 2ax+b=0 \iff x=-\frac{b}{2a}$ this is the critical point

now taking the 2nd derivative we get

$\displaystyle f''(x)=a \mbox{ but } a > 0$ so the graph is concave up

so the graph is decreasing on $\displaystyle (\infty,-\frac{b}{2a})$

and increasing on $\displaystyle (-\frac{b}{2a},\infty)$

- Apr 20th 2008, 05:02 PM #3