updated calc iii notes

content/calculus/function-of-several-variables.md

@@ -44,3 +44,19 @@ Theorem:
 \text{If } D(a,b) < 0 \text{, there exists a saddle point at }(a,b) \text{.}\\
 \text{If } D(a,b) = 0 \text{ the test is inconclusive.}
 \]
+
+## Lagrange Multipliers
+
+Theorem:
+\[
+\text{Let } f \text{ be a differentiable function in } R^2 \text{ that contains curve } C \text{ given by } g(x,y) = 0. \text{ Assume } f \text{ has a local extrema on } C \text{ at point } P(a,b). \text{ Then } \nabla f(a,b) \text{ is orthogonal to the tangent line of } C \text{ at } P. \text{ Assuming } \nabla g(a,b) \neq 0, \text{ then there is a real number } \lambda \text{, or lagrange multiplier, such that } \nabla f(a,b) = \lambda \nabla g(a,b).
+\]
+
+To find absolute extremas using lagrange multipliers:
+
+1. Find the gradient of \(f\)
+2. Find the gradient of \(g\)
+3. Set \(\nabla f(x,y) = \lambda \nabla g(x,y)\)
+4. Solve the system for \(x\) and \(y\) of \(\nabla f(x,y) = \lambda \nabla g(x,y)\) and \(g(x,y) = 0\).
+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.
+

layouts/_partials/math.html

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