I need to generate a function or possible function from known extrema. All my searching on this topic only tells how to find the extrema of a known function.
For example, given that $f(x)$ has a local minimum at $x = -3$ and a local maximum at $x = 3$, how would I go about finding a function that meets that criteria?
What if I also needed the function to be decreasing at $x = -5$?
Assuming your function is a polynomial then the given information speaks much of the derivative. The turning points of the curve are always zeros of the derivative, and hence, $x=-3$ and $ x=3$ are zeros of the derivative. We can write:
$$ P'(x) = (x- (-3) ) (x-3) J(x)$$
Or,
$$ P'(x) = (x^2 -9) J(x)$$
Where $P'(x)$ is the derivative our polynomial and $J(x)$ is the other factors lumped as a function. Now, since a local minima occurs at $x=-3$, $P''(x)$ must be positive at $ x=-3$ ,and similarly, it must be positive at $x=3$
Finally, if the function is decreasing at $x=-5$ that tells ut that $ P'(-5)$ is a negative quantity.
Hope this helps!!