Find the absolute maximum and absolute minimum values of z on D where z = f(x, y) = 2x3 + y4 and D = {(x, y)|x2 + y2 ≤ 1}. D is the boundary and interior of the unit circle.
1. Sketch D. What is the equation of the boundary of D?
2. Find the critical points of f.
3. Input the critical points into f.
4. Now it is time to input the boundary into f. The goal is to eliminate one of the variables of f. The easiest way is to do this is to isolate y2 in the boundary equation and input this into f. Do this. Write down your new function that is just in terms of x. Even though we have eliminated the ys from t we will still refer to this function as f.
5. Now we need to maximize and minimize f using Calculus I. First, what are the critical points off?
6. Input the critical points into f Record the z-values and the (x, y) locations.
7. Now we have to think about the boundary in terms of x. What is the largest and smallest x-values we can input into x?
8. Lastly, input the boundary points in terms of x into f. Record the z-values and their locations.
9. Now, compare all of your z-values. Give the absolute max and its location(s). Give the absolute min and its location(s).