First Derivative Test (function behavior)
f'(c)=0 –> x=c is a critical point of f(x)f'(x)>0 for all x in interval (a.b) –> f(x) is increasing on [a,b]f'(x)<0 for all x in interval (a,b) --> f(x) is decreasing on [a,b]f'(x)>0 for xc –> f(c) is a maximumf'(x)<0 for x0 for x>c –> f(c) is a minimum
The First Derivative Test is a method for analyzing the behavior of a function at critical points. A critical point of a function is a point where the derivative is either zero or undefined.
To use the First Derivative Test to determine the behavior of a function at a critical point, follow these steps:
1. Find the critical points of the function by setting the derivative equal to zero and solving for x.
2. Determine the sign of the derivative to the left and right of each critical point. You can do this by plugging in a value slightly less than the critical point into the derivative to find the sign to the left, and plugging in a value slightly greater than the critical point into the derivative to find the sign to the right.
3. Use the sign of the derivative to determine the behavior of the function at each critical point. If the derivative changes sign from negative to positive at a critical point, then the function has a local minimum at that point. If the derivative changes sign from positive to negative, then the function has a local maximum at that point. If the derivative does not change sign at a critical point, then the behavior of the function at that point is unclear and further analysis may be required.
The First Derivative Test can be a useful tool for analyzing the behavior of functions at critical points, but it is important to remember that it only provides information about local extrema. To determine the behavior of a function over its entire domain, additional information is needed.
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The First Derivative Test is a method for analyzing the behavior of a function at critical points. A critical point of a function is a point where the derivative is either zero or undefined.
To use the First Derivative Test to determine the behavior of a function at a critical point, follow these steps:
1. Find the critical points of the function by setting the derivative equal to zero and solving for x.
2. Determine the sign of the derivative to the left and right of each critical point. You can do this by plugging in a value slightly less than the critical point into the derivative to find the sign to the left, and plugging in a value slightly greater than the critical point into the derivative to find the sign to the right.
3. Use the sign of the derivative to determine the behavior of the function at each critical point. If the derivative changes sign from negative to positive at a critical point, then the function has a local minimum at that point. If the derivative changes sign from positive to negative, then the function has a local maximum at that point. If the derivative does not change sign at a critical point, then the behavior of the function at that point is unclear and further analysis may be required.
The First Derivative Test can be a useful tool for analyzing the behavior of functions at critical points, but it is important to remember that it only provides information about local extrema. To determine the behavior of a function over its entire domain, additional information is needed.
More Answers:
Master Optimization Problems For Math: Techniques, Constraints, And Decision Variables ExplainedMastering The Second Derivative Test: Analyzing Function Behavior And Critical Points
Finding Absolute Extrema Of Functions: Step-By-Step Guide With Example