Number of critical point for y=f(x) for x ∈[0,2] (a) 0 (b) 1...

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Key Concepts

Critical Point

A critical point of a function is a point in its domain where the derivative is zero or does not exist. These points are significant because they are potential locations of local maxima, local minima, or points of inflection, and play a crucial role in the analysis of a function’s behavior.

Derivative

The derivative of a function provides the rate at which the function is changing at any given point. It is the fundamental tool used to determine where a function has horizontal tangents (indicating potential extrema) or where the function’s behavior changes, which is why it is central to the identification of critical points.

Closed Interval

A closed interval is a set of numbers that includes its end points. When analyzing functions over closed intervals, one must consider the endpoints in the search for extrema, as the extreme value theorem guarantees that a continuous function on a closed interval will attain its maximum and minimum values at some points within the interval, even if those points are at the boundaries.

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