Critical Points In Calculus: Identifying Local Extremes And Inflection Points

f(x) has a critical point when

f’ = 0, f’ is undefined

A critical point of a function f(x) is a point x=c where the derivative of f(x) is either 0 or undefined. Mathematically, we can say that a function f(x) has a critical point at x=c if one of the following conditions hold:

1) f'(c) = 0
This condition means that the slope of the tangent line to the curve of f(x) at x=c is zero. In other words, the function is neither increasing nor decreasing at that point.

2) f'(c) is undefined
This condition means that the function has a vertical tangent or a cusp at x=c. In other words, the function is changing direction abruptly at that point.

At a critical point, the function may have a local maximum, a local minimum or an inflection point depending on the behavior of the function around that point. A local maximum occurs if the function is increasing before the critical point and decreasing after the critical point. A local minimum occurs if the function is decreasing before the critical point and increasing after the critical point. An inflection point occurs if the function changes concavity at the critical point.

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