\( \newcommand{\sech}{\operatorname{sech}} \) \( \newcommand{\inverse}[1]{#1^\leftarrow} \) \( \newcommand{\<}{\langle} \) \( \newcommand{\>}{\rangle} \) \( \newcommand{\vect}{\mathbf} \) \( \newcommand{\veci}{\mathbf{\hat ı}} \) \( \newcommand{\vecj}{\mathbf{\hat ȷ}} \) \( \newcommand{\veck}{\mathbf{\hat k}} \) \( \newcommand{\curl}{\operatorname{curl}\,} \) \( \newcommand{\dv}{\operatorname{div}\,} \) \( \newcommand{\detThree}[9]{ \operatorname{det}\left( \begin{array}{c c c} #1 & #2 & #3 \\ #4 & #5 & #6 \\ #7 & #8 & #9 \end{array} \right) } \) \( \newcommand{\detTwo}[4]{ \operatorname{det}\left( \begin{array}{c c} #1 & #2 \\ #3 & #4 \end{array} \right) } \)

MA 227 Standard S07

Optimization
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At the end of the course, each student should be able to…

  • S07: Optimiz. Use the first-derivative test and Lagrange multipliers to optimize a real-valued multivariable function.

S07: Optimization

  • A continuous function of multiple domains restricted to a closed and bounded domain attains a maximum and minimum value.
    • If the function is differentiable, these values must occur either at interior critical points where \(\nabla f = \vect 0\) or on boundary points.
    • So to optimize a function, find all the critical points in the interior, then find critical points on the boundary, and finally check any corner points. All of these points should be tested in the function, and the largest and smallest are the maximum and minimal values of the function.
  • The method of Lagrange Multipliers provides another method of finding where a function is optimized.
    • If \(P_0\) optimizes the function \(f(P)\) restricted to the domain \(g(P)=k\), then the curves/surfaces \(f(P)=f(P_0)\) and \(g(P)=k\) must have parallel normal vectors. In particular, \(\nabla f=\lambda\nabla g\) for some Lagrange multiplier \(\lambda\).
    • Thus if \(f(P)\) may be optimized on the domain \(g(P)=k\), the optimizing point \(P_0\) may be found by solving the system of equations given by \(\nabla f=\lambda\nabla g\) and \(g(P)=k\).

Textbook References

  • University Calculus: Early Transcendentals (3rd Ed)
    • 13.7 (exercises 31-38)
    • 13.8 (exercises 1-14, 17-28)