f has a local (relative) maximum at x=a
f(a) is greater than or equal to every other y-value in some interval containing x=a
If function f has a local (relative) maximum at x=a, it means that there exists some open interval (a-h, a+h) around x=a, such that f(x)≤f(a) for all values of x within this interval.
In other words, the function f reaches its peak value at x=a, and no nearby point within the interval can have a higher value than f(a).
The slope of the function at x=a is also zero or undefined. If the slope is zero, it means that the function is horizontal at that point, and if the slope is undefined, it means that the function has a vertical tangent line at that point.
We should note that having a local maximum at x=a does not necessarily mean that it is a global (absolute) maximum. There may be other points in the domain of the function where it attains a higher value than f(a).
To determine whether the local maximum at x=a is a global maximum, we need to examine the behavior of the function at the endpoints of the domain (if it is bounded) or at infinity. If the function is unbounded and has no endpoint, we need to consider its limit at infinity.
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