How can I find the maximum point on a function without using its derivative?

[brydustin]

I have a function with one and only one local/global maxium... (i.e. half the function has positive slope, half the function has negative slope). And I want to find the maximum point on the function. How can I find the function's max?

I was thinking of turning the function into its derivative and using Newton's Method for finding the zero... are there better ways?

 

[HallsofIvy]

Yes, that will work. Another way, that will converge slower does not require finding the derivative (and the second derivative to use Newton's method) is this:

Choose two points, p and q, and an interval length, $\delta$. If f(p)> f(q), choose a new point, r, a distance $\delta$ beyond p (opposite the direction from p to q. If f(q)> f(p), reverse p and q). If f(r)> f(p), repeat. If f(r)< f(p) reverse direction and divide $\delta$ by 2 to halve the interval length.

Since this question has nothing to do with differential equations, I am moving it to "Calculus and Analysis".