Understanding Derivatives in Optimization

[trim girder]

Can someone briefly explain why we take dy/dx=0 in optimization problems? Here’s how I understand it so far:
Taking DxY in an optimization problem is similar to taking DxY when finding the local extrema in a graph. When say a curve goes from - to 0 to + (concave up) it changes direction from - to +. In optimization, taking DxY = 0 essentially illustrates that at DxY=0 / DxY=local extrema, I’m imagining a line moving at a rate DxY until it starts approaching 0, and when it reaches 0 the line changes direction.
Is that way of thinking correct or am I misunderstanding? Thanks

[sharp crane]

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[humble stump]

I mean, apart from me not recognizing your notation, you're pretty spot-on. dy/dx = 0 indicates a local extremum, which indicates that y is minimized/maximized at that value of x.

Remember, dy/dx is the slope of the line tangent to the function at the point x. Therefore, dy/dx = 0 indicates that the tangent line at that point is a horizontal line. This doesn't always indicate a local extremum, but it makes it a good point to investigate if you're looking for one.

[trim girder]

Ah right I have to do the second derivative test/interval chart to be sure its a local extremum. Just to be sure, the “a line moves at rate dy/dx and it starts slowing down as it reaches 0, and when it reaches 0 it changes direction” part is generally correct right? Cuz ive been getting mixed answers on that last part, thanks!

[crisp leaf]

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