Question

Find and classify the critical points of these functions (that is, are they local maxima, minima, saddle points, or points where the function is not differentiable)

(a) h(x, y) = (12-2) (b) k(x,y) = sin(I) cos(y), with the domain {(1,y) |+ y2 < 4}.

Answer 1

Since his exponential function, his differentiable (-x) К(,) : e а. every where Wъ /h (, ") - ( з əh (x, y) , ah. ЭР -Гу»)) ә gh (2) - (~-~~) e С 2) ә • 29) ди ә) (+»). ? ?е gh (ъ)) : [") г) (-21 е (For criticul points ofh) **)-(0,9) y 2ъ ay 2 2 e *to other 7h (7,9)= (99) 2) (+ 2) = (-), A. (0,0) is the only critical point of hi дун (":"") 4-") 2) = -4». *by) = (-1») е әһә əh (7) әрәх әх ә) ә (5) - Рүбүз фғІ - 2e (ә»") ах с (-2») ә?, О" ) (4ungle (У х") + 24 е а, 15*»") Хе • З ayr (45h+2) (о?» 1)

K(n,y)= sin(n) (y) of an syn 215r.n) e 21gh-x) Abrhyne (ahh art (4222) (49h+2) [lsamy 1633*+ 8y *- *** +4] elykn) (8yr-surtu) e 2 y 2 ) e - A (0,0) - he of h(n"), .. domain At the per critical point (ou). 2 (ywany yo. (90) is a suddle point (6 with the domain Da 2. Guin) lärtyr<44 Kl is differentiable the ək TK (2,4) 2 all en x eusya eus y = sinn siny =o WA) lak (2,7) say an FOY given (6,»> ) = 6,0) - sinn siny) = (0,0) yen 10 it eus x=0 then, sinnt erst = sin =) na (for ne te 0 to, L Cu & sin :. as Flye to MIL a D NK andy ayan siny esa (2.0) (ED) ark sirin asy dyn a lay ^? easy ark dar ark age a sina sina esry. 11 as a sing sin' (tj ces "o) wash (1) sin(0) A (119)? -1 <0 11

m maximum (7,0) = -f co an relative mexim K(2,5) ( has a v naty 24} at 1,0) € { (2, 9) A Cone 2 ولن 22g으 y casryt sin?y =) sin yat to i to, (fir y = 2 sin 22 0 (from (A)) 2. nao, When is a (0) is critical point for khim) 163 =tso (Alemania este ning mica saddle point at (o !) .. klart has a