In this discussion, we do the opposite. We start with what we want the graph of a function to look like and then try to find a function that has those properties. This type of problem is useful in design when we have a target end result $($shape of the curve$)$ and need to find a way to build something $($a function$)$ that leads to that target result. For example, this could be used in noise reduction where engineers remove static from sound.

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For this week's discussion, you are asked to generate a continuous and differentiable function f$($x$)$fx with the following properties:

f$($x$)$ is decreasing at x$=-6$

f$($x$)$ has a local minimum at x$=-2$

f$($x$)$ has a local maximum at x$=2$

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Hints:

Use calculus!

Before specifying a function f$($x$),$ first determine requirements for its derivative f$($x$).$ For example, one of the requirements is that f$(2)=0.$

If you want to find a function g$($x$)$gx such that g$(9)=0$and g$(8)=0,$ then you could try$$g$($x$)=($x$+9)($x$8).$

If you have a possible function for$$f$($x$),$ then use the techniques in Indefinite Integrals this Module to try a possible$$f$($x$).$

f$($x$)=$