13.2 Critical points and the First Derivative Test
2 min read•july 22, 2024
Critical points are key to understanding a function's behavior. They occur where the derivative is zero or undefined, indicating potential turning points or discontinuities in the graph.
The helps determine if these points are local maxima, minima, or neither. By examining how the derivative's sign changes around critical points, we can identify peaks, valleys, and saddle points.
Critical Points and the First Derivative Test
Critical points of functions
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Top images from around the web for Critical points of functions
StationaryPoints | Wolfram Function Repository View original
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StationaryPoints | Wolfram Function Repository View original
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Points where the derivative is zero (stationary points) or undefined (non-differentiable points)
Stationary points occur when the is horizontal (f′(c)=0)
Non-differentiable points arise from vertical tangents, cusps, or discontinuities (f′(c) is undefined)
Found by setting f′(x)=0 and solving for x, then checking for any x-values that make f′(x) undefined
First derivative test conditions
Determines the nature of critical points and relative extrema for a continuous function f on an open interval containing the critical point c
If f′ changes from positive to negative at c, then f(c) is a (peak)
If f′ changes from negative to positive at c, then f(c) is a (valley)
If f′ does not change sign at c, then f(c) is neither a local maximum nor a local minimum ( or )
Nature of critical points
Apply the first derivative test to determine if a critical point is a local maximum, local minimum, or neither
Find the critical points of the function by setting f′(x)=0 and solving for x, and identifying any x-values that make f′(x) undefined
Evaluate the sign of f′(x) on the left and right sides of each critical point using test points
If the sign changes from positive to negative, the critical point is a local maximum
If the sign changes from negative to positive, the critical point is a local minimum
If the sign does not change, the critical point is neither a local maximum nor a local minimum
Relative extrema using derivatives
Relative extrema are the local maxima and minima of a function
Found by applying the first derivative test to each critical point
If f′ changes from positive to negative at a critical point, it is a local maximum
If f′ changes from negative to positive at a critical point, it is a local minimum
The y-coordinates of the local maxima and minima are the relative maximum and minimum values of the function, respectively