Local minima and maxima of multivariable functions
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Intros
Lessons
- Local Minima & Maxima of Multivariable Functions Overview:
- Definition of Critical Points
- Critical points for 1 variable
- Critical points for 2 variables
- An example
- Types of Critical Points
- Local maximum
- Local minimum
- Saddle point
- Classifying Critical Points
- Calculate D
- D>0 and fxx(x0,y0)>0→ local minimum
- D>0 and fxx(x0,y0)<0→ local maximum
- D<0→ saddle point
- D<0→ saddle point
- D=0→ failed to classify
- An example
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Examples
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Topic Notes
Notes:
Definition of Critical Points
Types of Critical Points
Classifying Critical Points
Suppose that (x0,y0) is a critical point of f(x,y). To see whether it's a local maximum, or local minimum, or saddle point, we compute the following:
Definition of Critical Points
Recall for 1 variable functions, a critical point occurs at a point x0 if f′(x0)=0 or f′(x0) does not exist. The concept is the same for 2 variable functions, except we must modify a few things.
The point (x0,y0) is a critical point of f(x,y) if one of the following is true:
- fx(x0,y0)=0, fy(x0,y0)=0
- fx(x0,y0) and/or fy(x0,y0) does not exist
Types of Critical Points
There are 3 types of critical points:
- Local Maximum: occurs when f(x0,y0)≥f(x,y) for all points (x,y) that is around (x0,y0). In other words, it's the biggest value of the function around it's region.
- Local Minimum: occurs when f(x0,y0)≤f(x,y) for all points (x,y) that is around (x0,y0). In other words, it's the smallest value of the function around it's region.
- Saddle point: neither a local minimum or local maximum.
Classifying Critical Points
Suppose that (x0,y0) is a critical point of f(x,y). To see whether it's a local maximum, or local minimum, or saddle point, we compute the following:
D=fxx(x0,y0)⋅fyy(x0,y0)−[fxy(x0,y0)]2
If:
- D>0 and fxx(x0,y0)>0, then it is a local minimum
- D>0 and fxx(x0,y0)<0, then it is a local maximum
- D<0, then it is a saddle point
- D=0, then it could be any of the 3 types. Need to use other techniques to classify it.
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