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17 Cards in this Set
- Front
- Back
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Local minimum |
In a scalar function, x₀ is a local minimum of f if there is a neighborhood V of x₀ such that for all points x in V, f(x) ≥f(x₀) |
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Local Maximum |
x₀ is a local maximum if there is a neighborhood V of x₀ such that for all points x in V, f(x)≤f(x₀) |
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Local/relative extremum |
A maximum or minimum |
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Critical point |
A point x₀ is a critical point of f if either f is not differentiable at x₀, or if it is, Df(x₀)=0 |
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Saddle point |
A critical point that is not a local extremum |
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First derivative test for local extrema |
If the function is open and differentiable and x₀ is a local extremum, then Df(x₀)=0; that is, x₀ is a critical point of f |
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Hessian of f at x₀ |
Suppose that f: U⊂Rⁿ→R has second-order continuous derivatives (δ²f/δxiδxj)(x₀), for i,j=1,...,n, the hessian of f at x₀ is defined by Hf(x₀)(h)= ½∑δ²f/δxiδxj (x₀) hihj |
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Second derivative test for local extrema |
If f:U⊂Rⁿ→R is of class C³, x₀ is a critical point of f, and the hessian Hf(x₀) is positive definite, then x₀ is a relative minimum of f. Similarly if Hf(x₀) is negative-definite, then x₀ is a relative maximum |
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Second Derivative Maximum-Minimum Test for functions of Two Variables |
Let f(x,y) be of class C³. A point (x₀,y₀) is a local minimum of f provided 3 conditions hold 1. δf/δx(x₀,y₀) = δf/δy(x₀,y₀) = 0 2. δ²f/δx²(x₀,y₀)>0 3. D= (δ²f/δx²) (δ²f/δy²) - (δ²f/δxδy)²>0 at (x₀,y₀) Where D is the discriminant of the Hessian If condition 2 is<0, then we have a local maximum |
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What happens if D<0 in the Second Derivative Maximum-Minimum Test for functions of Two Variables? |
We have a saddle point |
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Nondegenerate critical point |
Critical points for which D≠0 |
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Degenerate critical points |
Critical points where D=0 |
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Absolute maximum |
f(x)≤f(x₀) |
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Absolute minimum |
f(x)≥f(x₀) |
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Bounded |
A set D∈Rⁿ is said to be bounded if there is a number M>0 such that ‖x‖<M for all x∈D |
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Closed |
A set is closed if it contains all its boundary points |
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Strategy for Finding the Absolute Maxima and Minima on a Region with Boundary |
Let f be a continuous function of two variables defined on a closed and bounded region D in R², which is bounded by a smooth closed curve. To find the absolute maximum and minimum of f on D: 1. locate all critical points for f in U 2. Find the maximum and minimum of f viewed as a function only on δU 3. Compute the value of f at all these critical points. 4. Compare all these values and select the largest and smallest |
