Math 411 spring 2013 boyle final exam. Points in rn are column vectors (even if they are typed horizontally). Given part (b), use an appropriate inequality to prove that if u is constrained to have some. Xed positive length r then duf (p) is maximized when u is a positive multiple of f (p): (30 pts) de ne f : r2 r by setting f (x, y) = ex+xy+y2. Compute n(r) for n = 3: (50 pts) let e = {(x, y) r2 : x 0, y 0 and x + y = 1}. De ne f (x, y) = 8x2 + y2. Find all points in e at which f assumes a maximum or minimum. Justify your claims that f assumes extreme values at these points and at no other points: (40 pts) for a pair of real numbers c and d, consider the system of equa- tions x + x2 + ex2y2.
