63 lines
2.8 KiB
Markdown
63 lines
2.8 KiB
Markdown
+++
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date = '2026-03-02T19:00:00-06:00'
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draft = false
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title = 'Function of Several Variables'
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layout = 'chapter'
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type = 'book'
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tags = 'calculus'
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chapterno = 15
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+++
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## Max/Min Problems
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-------------------
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Definition:
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\[
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\text{Suppose } (a,b) \text{ is a point in region} R \text{ on which } f \text{ is defined and there is an open disk centered at } (a,b) \text{.}\\\\
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\text{If: } f(x,y) \leq f(a,b) \text{, then } f(a,b) \text{ is a local max.}\\
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\text{If: } f(x,y) \geq f(a,b) \text{, then } f(a,b) \text{ is a local min.}
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\]
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Theorem:
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\[
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\text{If } f \text{ has a local max or min value at } (a,b) \text{ and the partial derivatives } f_x \text{ and } f_y \text{ exist at } (a,b) \text{, then } f_{x}(a,b) = f_{y}(a,b) = 0 \text{.}
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\]
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If you are at \((a,b)\) and every where you look, the value next to you is higher, you are at a local minimum position. If everywhere you look, the value is lower, you are at a local maximum position.
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Critical points are identified by finding where the partial derivatives of each variable in the multi-variate function equals zero.
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Definition:
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\[
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\text{ A critical point in } f \text{ is located as an interior point } (a,b) \text{ if either:}\\\\
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f_{x}(a,b) = f_{y}(a,b) = 0\\
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f_{x} \text{ or } f{y} \text{ does not exist at } (a,b)
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\]
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If you are able to find the critical points of \(f\) then you can use the _Second Partial Derivative Test_ to determine the maximum and minimum values.
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Theorem:
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\[
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\text{If the second partial derivatives of } f \text{ are continuous throughout an open disk centered at } (a,b) \text{. Let } D(x,y) = f_{xx}(x,y)f_{yy}(x,y) - (f_{xy}(x,y))^{2} \text{.}\\\\
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\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local max at } (a,b) \text{.}\\
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\text{If } D(a,b) > 0 \text{ and } f_{xx}(a,b) < 0 \text{, there exists a local min at } (a,b) \text{.}\\
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\text{If } D(a,b) < 0 \text{, there exists a saddle point at }(a,b) \text{.}\\
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\text{If } D(a,b) = 0 \text{ the test is inconclusive.}
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\]
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## Lagrange Multipliers
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Theorem:
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\[
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\text{Let } f \text{ be a differentiable function in } R^2 \text{ that contains curve } C \text{ given by } g(x,y) = 0. \text{ Assume } f \text{ has a local extrema on } C \text{ at point } P(a,b). \text{ Then } \nabla f(a,b) \text{ is orthogonal to the tangent line of } C \text{ at } P. \text{ Assuming } \nabla g(a,b) \neq 0, \text{ then there is a real number } \lambda \text{, or lagrange multiplier, such that } \nabla f(a,b) = \lambda \nabla g(a,b).
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\]
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To find absolute extremas using lagrange multipliers:
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1. Find the gradient of \(f\)
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2. Find the gradient of \(g\)
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3. Set \(\nabla f(x,y) = \lambda \nabla g(x,y)\)
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4. Solve the system for \(x\) and \(y\) of \(\nabla f(x,y) = \lambda \nabla g(x,y)\) and \(g(x,y) = 0\).
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5. Evaluate the points you've found in \(f(x,y)\). The largest value is the absolute maximum. The smallest value is the absolute minimum.
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