Local ExtremaDo you remember in Calculus 1 & 2, we

sometimes found points where both f'(x) and f"(x) were zero?

What did this mean?? What is the corresponding condition for f(x,y)??

Consider Ex 3 pg 178.

Local ExtremaDo you remember in Calculus 1 & 2, we

sometimes found points where both f'(x) and f"(x) were zero?

Define Saddle Point:

A function, f, has a saddle point at Po if Po is a critical point and within any distance of Po no matter how small, there are points P1 and P2 with

f(P1) > f(Po) and f(P2) > f(Po).

See Fig. 14.8

Local ExtremaDo you remember in Calculus 1 & 2, we

sometimes found points where both f'(x) and f"(x) were zero?

Just as in the calculus of 1 variable, we had to determine if critical points were maxima, minima, or neither (inflection points).

For functions of 2 variables, we must determine if critical points are maxima, minima, or neither (saddle points)!

Local ExtremaRemember also from one variable

calculus, that we could classify critical points as max, min, or neither based on the behavior of the 2nd derivative (2nd Deriv Test).

For functions of 2 variables, we have an analogous but (as expected) more complicated 2nd Derivative Test.

Local ExtremaSecond Derivative Test for z=f(x,y):

If (x0,yo) is a critical point (grad f (x0,yo) = 0) and

Then

• if D>0 and fxx(x0,yo) > 0, f has a local min at x0,yo.

• if D>0 and fxx(x0,yo) < 0, f has a local max at x0,yo.

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

Local ExtremaSecond Derivative Test for z=f(x,y):

If (x0,yo) is a critical point (grad f (x0,yo) = 0) and

Then

• if D>0, f has a saddle point at x0,yo.

• if D=0, anything is possible.

See pg 182. Work thru Ex 5 pg 183

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

Local ExtremaSecond Derivative Test for z=f(x,y):

See pg 182. Work thru Ex 6 pg 183

You might also recall doing the algebraic type analysis for 1 variable extrema when f"(xo)=0!

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

Local ExtremaSecond Derivative

Test for z=f(x,y):

Consider

What are the signs of fxx and fyy?

What is fxy? D?

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

2y2xy)f(x,

Local ExtremaSecond Derivative

Test for z=f(x,y):

Consider

What are the signs of fxx , fyy?

What is fxy? D?

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

2y2xy)f(x,

Local ExtremaSecond Derivative

Test for z=f(x,y):

Note that if surface is concave up in 1 direction and concave down in the other, we expect a saddle point.

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

2y2xy)f(x,

Local ExtremaSecond Derivative

Test for z=f(x,y):

Consider

What are the signs of fxx , fyy?

What is fxy? D?

2ooxyooyyooxx ))y,x(f()y,x(f)y,x(fD

2y2xy)f(x,

OptimizationAgain as we did in 1 variable Calculus,

we can use the methods for finding maxima and minima to solve 2 variable optimization problems.

To start, we distinguish between global and local extrema. What is the difference?? Can they be the same??

OptimizationAgain as we did in 1 variable Calculus,

we can use the methods for finding maxima and minima to solve 2 variable optimization problems.

Do you remember how you solved optimization problems for 1 variable calculus????

OptimizationTo Solve the Unconstrained

Optimization Problem:

1) Find the critical points.

2) Investigate whether critical points give global maxima or minima.

Work thru Ex 2 pg 187

Examples: pg 192 #3