Question

Part 2 nts) 3. Let x = e' cos 0, y = e' sin 0, and suppose f: R2 + R is a C f(e cos 0,e" sin 8). Show that function. Let g(r, 0) = 09 (,0) + 3 (1, 0) = e) (no elle" cos ,esin ) + face" cos 8, e' sine)). nts) 4. Let S = {(x,y) € R2: xy #0}, and define f : S R by if y > 0 f(x,y) = x 10 otherwise Show that for any p = (a, b) with a > 0 and b > 0, there is a point q € S such that if(p) -f(9) > Vf(c). (p - 9) for every c ES. uts) 5. Find and classify (as local minima, local maxima, or saddle points) all critical points of f(x,y) = ***+ +82? + 6xy +

dk how to do question 3 and 4 . help !!

Answer 1

3) xre cos, yre sin Orx = e' cos@=x, = d sin 6 = y ......(1) oc ag ag aa Ox=-sin 0=-y. cos&=x.......... 8(1,0)=f(x,y) ag _ af ac af əy _ .af ... of By (1) ar ac ar ər ar ac Ox of a cady a 7+ 12+ arlar ) or a ər ôc ar (ax) ar əy 0,82 (alaray äxt võttää jät Yalójá = y + y + x2 + y2 By (1.....(in) əg_Of or of dy- tre By (m) 20 odezy ãe-Max By (ü) of of Oy of Of Ox Ox of foy **@y J 20 x x 2020 **@y? 20 -- - + y + xosby (m1........(iv) Adding (i) and (iv)We get a'g og in O1 O2 + or? =e no a toy? 002 I ac ar ôx² əv² an (alf of -e(cos? +sin bzir tv- 6 = 2 391,93 4 3 570 = ( 01.) * L.:") 1819), PEROS-o se con0,e’sino). Se cos 6, e' sino) Hence, Proved