Extrema

Today we reviewed extrema and applied it to functions with several variables. Finding extrema on functions with variables uses a similar technique; it is imperative that we test not only the critical points but all boundaries given for the problem.

Local extrema: A function value that is greater than/ less than or equal to all the nearby functions

Finding extrema on closed intervals:
Extreme Value Theorem: If f(x) is continuous on [a,b], then f must take a maximum value and a minimum value on [a, b]

Procedure:
1) Find and test critical points (where f'(x)=0 or f'(x)=undefined)
2) Check the endpoints/boundaries. The boundaries can be checked by eliminated one variable (ex. x=1, y=0, ...etc)

Remember to set up a chart (x,y) | f in order to determine clearly which values are at a maximum and which values are at a minimum.

Boundary examples:
1) 0 < x < 3 and 0 < y < 3
Draw a graph of the boundaries and label them accordingly. When testing, use the intervals.
2) Triangular region whose vectors are (2,0) (0,1) and (1, 2)
Similar approach to rectangular boundaries. Make sure you label D1 D2 and D3 and eliminate variables in order to solve the equation.
3) Circle
If given an equation F(x,y) = x^2+y^2-4y R={(x,y)| x^2+y^2<16}, you can substitute 16 for x^2+y^2 in order to turn the function into one variable (y). You can then check the endpoints (-4 and 4) as well as the critical points.