You need to be aware that some functions will have two or more critical points, and you have to use other tests to determine which one is optimal in terms of the way the problem was set forth.
The rate of change at these exact moments is zero, so we hunt for optimization points by finding the derivative and then determining where the derivative is equal to zero (the “critical points”). It’s the top of a frown-shaped curve or the bottom of a smile-shaped curve. The tipping point is where the derivative starts to go in the other direction. As long as the rate of change is in the “good” direction (which may be up or down, depending on what you’re optimizing), we keep going. We find those tipping points by looking at the derivative, which is the rate at which something is changing. Or it’s getting smaller, then it starts to get bigger. It’s getting bigger, then it starts to get smaller.
Something is getting better up to a point, and then it starts to get worse. Optimization problems have to do with finding a tipping point. I don’t know how much help you can get from a brief reply here, but I’ll offer some comments for what they’re worth.
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