Contour Plots and Critical Points

Part 3: Locating the Critical Points

1. Enter the first of your approximate critical points -- the one that should lead to a maximum or minimum value of f -- as (x₀, y₀). [Use your 2SD approximation from step 4 of Part 1 -- not a more accurate approximation if you found one in step 6 of Part 1.] Be sure that Q is defined as the second-dgree Taylor approximation at this point.

2. Since it is difficult to work with the partial derivatives of f, we will work with the partial derivatives of Q instead. Solve the system of equations
   

   Define the resulting solution point to be (x₁, y₁).

3. Calculate the values of f, f_x, and f_y at (x₁, y₁). Replace (x₀, y₀) by (x₁, y₁), and redefine Q to be the second-order Taylor approximation at this new point. Then study the simultaneous contour plot of f and this new approximation Q.

4. Repeat the process of steps 2 and 3 -- solving for the critical point of Q, and then redefining Q. Continue until you are sure that you have coordinates for the critical point that are accurate to five significant digits (5SD). Record your result.

5. Have you found a local maximum point or a local minimum point? How can you tell? (Hint: Use the second derivative test.)

6. Repeat the process in steps 2 through 4 for your second estimated critical point, the one that should lead to a saddle point. [Again, start with a 2SD estimate.]

7. Have you in fact located a saddle point? How can you tell?

8. Go back to your contour plot of f in Part 1, and approximate a third critical point to two significant digits. If you have already found a local maximum, find a local minimum. If you have found a local minimum, find a local maximum. Then use the second-degree Taylor approximations to find the coordinates of this point to five significant digits.

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