maximizing/minimizing a smooth real-valued function. Consider the map

f : R3 > R given by f(m,y,z) : 21.312 _ 4133141132 + 252 22. (a) Since this map is smooth, a critical point of f is a point (as, y, z) E R3 such that Vf(:l:, y, z) = 0. Our map f has exactlythree critical points. Find those three critical points and classify them (ire, determine ifthey are local minima, local maxima or saddle points). (b) As a function on the entire R3 , f need not have an absolute extremum. So, let's restrict it to a compact subset of R3. Let K be the portion of the plane 3/ = 0 inside the elliptic cylinder :02 + 42,2 = 4. Making 3; = O in the expression of f, we see that the restriction of our function f to the compact setK is the function F ; E ) R, given by F(w,z) = 562 + 7.2 22 where E = {(m, z) e R2 ; :32 + 422 g 4}. Find the maximum and the minimum values of ourfunction f on K. In other words,nd the maximum and the minimum values of the restricted function F on E. I suggest doing this in 3 steps like we did in class: Step 1) List the critical points of Fin int(E) = {ob,7.) 1:132 + 4.z2