can someone break it down barney style for me. what is the difference between relative max and min and local extrema. Why do I sometimes have both, but sometimes only max and min? whats the difference between absolute max and relative max? I'ts confusing me.

Slope sign changes can indicate a minimum or maximum. If the value of a maximum or minimum differs from the largest maximum or minimum for that equation, then it's a local max or min.

Consider a sine wave - it has an infinite number of mins and maxes. A straight sine wave has all maxes and mins of the same magnitude, but with a long axis variable, like a sine curve which damps the further from the y axis it gets, the peaks and troughs of its graph can vary, giving different values each max and min.  Overall, the equation will have one ultimate max and min, but many locals, all potentially with different values.

Slope changes, the change in sign of the NEXT derivative to determine curvature direction (peak v. trough).

Good ole' analysis.

Works even when the variables are matrices and tensors.

Absolute maximum/minimum can occur on the boundary of the domain.

Suppose your function is:

   y = x - (x^3)/6 + (x^5)/120

Suppose the allowed domain for X is -4 to +4 then
The absolute minimum is at X = -4 and Y = -2 (approximately)
The absolute maximum is at X = 4 and Y = 2 (approximately)
There are two relative maximums in that range, and two relative minimums in that range

Not a useful result in many cases.