Simple functions - zero of f(x)

• Sep 28th 2009, 03:32 AM
lantern4christ
Simple functions - zero of f(x)
I have precalculus in college. I am trying to figure out a definitive formula to answer these equations. I'm not sure if they have a formula or if you kind of just go with it. Help would be appreciated. This is a 2-step problem. (a) find the zero of the function. (b) find the min/max of the parabola.(Happy)
• Sep 28th 2009, 03:57 AM
Defunkt
Quote:

Originally Posted by lantern4christ
I have precalculus in college. I am trying to figure out a definitive formula to answer these equations. I'm not sure if they have a formula or if you kind of just go with it. Help would be appreciated. This is a 2-step problem. (a) find the zero of the function. (b) find the min/max of the parabola.(Happy)

To find the zero of a function $\displaystyle f(x)$, we simply need to solve $\displaystyle f(x) = 0$ with regards to $\displaystyle x$. That is, find $\displaystyle x_0$ such that $\displaystyle f(x_0) = 0$.

To find the min/max of a parabola, we solve $\displaystyle f'(x) = 0$ with regards to x. Then, for those values of x we find $\displaystyle f''(x)$. Say $\displaystyle f'(x_0)=0$. If $\displaystyle f''(x_0)>0$ then $\displaystyle (x_0,f(x_0))$ is a minimum point. If $\displaystyle f''(x_0)<0$, then $\displaystyle (x_0,f(x_0))$ is a maximum point.

Example:
Let $\displaystyle f(x) = x^2 - 9$. Find the zeroes of $\displaystyle f(x)$ and find its min/max points.

First we'll find the zeroes. Solve $\displaystyle x^2-9=0$:
$\displaystyle x^2-9 = 0 \Rightarrow x^2 = 9 \Rightarrow x = \pm 3$
So $\displaystyle f(3) = f(-3) = 0$.

Now we'll find the min/max points. First we find $\displaystyle f'(x)$:
$\displaystyle f'(x) = \frac{d}{dx}(x^2-9) = 2x$

Solve $\displaystyle f'(x) = 0$: $\displaystyle 2x = 0 \Rightarrow x = 0$
So $\displaystyle x=0$ is a critical point. Find $\displaystyle f''(x)$:
$\displaystyle f''(x) = \frac{d}{dx}(2x) = 2 \Rightarrow f''(0) = 2$

So $\displaystyle (0,f(0)) = (0,-9)$ is a minimum point, since $\displaystyle f''(0)>0$
• Sep 28th 2009, 04:21 AM
CaptainBlack
Quote:

Originally Posted by Defunkt
To find the min/max of a parabola, we solve $\displaystyle f'(x) = 0$ with regards to x.

Not in pre-calculus we don't, here we rely on the properties of the parabola (see this for more information).

CB
• Sep 28th 2009, 04:42 AM
Defunkt
Oh, er, my bad then :S