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Show that $f(x)=x^{2}+b x+c$ is decreasing on $\left(-\infty,-\frac{b}{2}\right)$ and increasing on $\left(-\frac{b}{2}, \infty\right)$

For $x>-\frac{b}{2},$ we have $f^{\prime}(x)>0,$ so $f$ is increasing on $\left(-\frac{b}{2}, \infty\right)$

Calculus 1 / AB

Chapter 4

APPLICATIONS OF THE DERIVATIVE

Section 3

The Mean Value Theorem and Monotonicity

Derivatives

Differentiation

Applications of the Derivative

Harvey Mudd College

University of Nottingham

Idaho State University

Lectures

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In mathematics, precalculu…

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Proof Use the definitions …

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01:22

If $f^{\prime}(x)>0$ fo…

03:16

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02:25

Let $f(x)=a x^{2}+b x+c,$ …

relax is equal to X squared plus B R X plus C is increasing when x is less than negative, be over to an increasing when X is greater than negative view or two. So what we can do here is that we know if a if the derivative of a function is positive, the original function is increasing and if it is negative, the original function is decreasing. So what we can do here is find Theodore effective of this function so we can use the power down rule here. So the derivative of X squared is two acts on a derivative of be access be and you derivative c as a constant. So it's zero. So what we can do here is we want to find out when the derivative will be positive, because that's when FX will be increasing. So we can say to expose B is greater than zero and then we can solve for X, and we see that X is greater, then be over too. Now we can find the interval where it is less than zero so that the original function is decreasing. So we get X is less than be over too and we see that thes two intervals that we get are the ones that we intended to prove and to be getting.

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