Section 15.2 Optimization

Recall from single variable calculus Let f be continuous on a closed, bounded interval, [a,b], then f(x) has both a global maximum and a global minimum on [a,b] To find the global maximum and global minimum, it is sufficient to check –The critical points of f on [a,b] –The endpoints of the interval

Definitions If f is a function defined on a region R then f has a global maximum on R at the point P 0 if f(P 0 ) ≥ f(P) for all points P in R f has a global minimum on R at the point P 0 if f(P 0 ) ≤ f(P) for all points P near R

Let R be a region in the xy-plane P is called an interior point Q is called a boundary point R P..Q.Q

R is closed if it contains all of its boundary points R is bounded if all of its points lie within a sufficiently large radius –It does not stretch to infinity in any direction Examples –The disk is closed and bounded –The first quadrant x ≥ 0, y ≥ 0 is closed but not bounded –The disk is bounded but not closed –The upper half of the plane, y > 0 is neither closed nor bounded

Extreme Value Theorem for Multivariable Functions If f(x,y) is a continuous function on a closed, bounded region, R, then f(x,y) has both a global maximum and global minimum on R Further, it is sufficient to look at the following –The critical points in the interior of R –The boundary points of R

Example Let be defined on R where R is given by the following diagram Find the global max and min L1L1 L2L2 L3L3 R 5 3

A little note about parametric curves To find parametric equations, we need to take into consideration where we are starting at and where we are ending at as we travel from point to point Let’s look at finding the parametric equations that will give us the triangle from the previous problem We will use them along with maple to get a graphical representation of what is going on

Example Optimize on R given below (1,0) r=3