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Exercise 11.12. For f(x, y, z) = x2 + y² + 2², consider points P (0, 0, 1) that lie on the surface S = {g(x, y, z) = 1} for g(x, y, z) = x² + y² + z and have the tangent plane to S at P equal to the tangent plane to the level set of f through P. Show that all such P lie on the level set f = 3/4, and that the collection of such P is a circle in the plane z = 1/2. Hint: two planes through a common point coincide exactly when normal directions to the plane coincide. (Be attentive to the possibility of vanishing for various coordinates at such a P.)

Exercise 10.4. Let f(x, y) = x² + 3xy + y², and let D be the region defined by y ≥ 1 and x² + y² ≤ 10. (This is the part of the disk x² + y² ≤ 10 on or above the line y = 1.) (a) Draw a picture of D, indicating any "corners" on its boundary. (b) Find the critical points of f in the interior. (c) Find the extrema of f on the boundary of D by describing both the bottom edge and circular arc in terms of x alone (making the analysis on both the bottom edge and circular arc a single-variable calculus problem). Combine with (b) to find the extrema of f on D (find both the extreme values and where these are attained).