$\newcommand{\dis}{\displaystyle} \newcommand{\m}{\hspace{1em}} \newcommand{\mm}{\hspace{2em}} \newcommand{\x}{\vspace*{1ex}} \newcommand{\xx}{\vspace*{2ex}} \let\limm\lim \renewcommand{\lim}{\dis\limm} \let\fracc\frac \renewcommand{\frac}{\dis\fracc} \let\summ\sum \renewcommand{\sum}{\dis\summ} \let\intt\int \renewcommand{\int}{\dis\intt} \newcommand{\sech}{\text{sech}} \newcommand{\csch}{\text{csch}} \newcommand{\Ln}{\text{Ln}} \newcommand{\p}{\partial} $

Lecture 16, 9/28/2022. This page is for Section 1 only.
ACMS 20550: Applied Mathematics Method I
Instructor: Bei Hu, b1hu@nd.edu, Hurley 174A

  1. Maximum and Minimum
    1. Review of single variable function $ y = f(x)$.
      1st derivative test: If $y=f(x)$ attains a local minimum or maximum at $x=a$, then $f'(a)=0$.

      2nd derivative test:
      1. If $f'(a)=0, \; f''(a)<0$,
        then $y =f(x)$ attains a local maximum at $x=a$.
      2. If $f'(a)=0, \; f''(a)>0$,
        then $y =f(x)$ attains a local minimum at $x=a$.
    2. Functions of two variables: $z =f(x,y)$.
      1st derivative test: If $z=f(x,y)$ attains a local minimum or maximum at $(x,y)=(a,b)$,
      then $\frac{\p f}{\p x}(a,b)=\frac{\p f}{\p y}(a,b)=0$.

      2nd derivative test (problem 2 on page 213): Assume $f_x(a,b)=f_y(a,b)=0,$
      1. If $f_{xx}>0, f_{yy}>0,$ and $f_{xx}f_{yy}> f_{xy}^2$,
        then $z =f(x,y)$ attains a local minimum at $(x,y)=(a,b)$.
      2. If $f_{xx}<0, f_{yy}<0,$ and $f_{xx}f_{yy}> f_{xy}^2$,
        then $z =f(x,y)$ attains a local maximum at $(x,y)=(a,b)$.
      3. If $f_{xx}f_{yy}< f_{xy}^2$, then $z =f(x,y)$ attains
        neither a local maximum nor a local maximum at $(x,y)=(a,b)$.
    3. Examples of optimization.
      Step 1: Set up the equations, using a picture if necessary
      Step 2: Eliminate extra variable(s).
      Step 3: Set the 1st order derivatives to zero and solve.