Question

GROUP WORK 1, SECTION 14.3 Clarifying Clairaut's Theorem Consider f (x, y, z) = x?cos (y + 2). 1. Why do we know that fyyxxx=0 without doing any computation? 2. Do we also know, without doing any computation, that Sxyz = 0? Why or why not? 3. Suppose that a = 3x + ay". Jy = bxy + 2y. S,(1, 1) = 3, and has continuous mixed second partial derivatives xy and fyx. (a) Find values for a and b and thus equations for fi and fy. HINT What does Clairaut's Theorem say about the mixed partial derivatives of a function? When does the theorem apply? (b) Can you find a function F(x, y) such that = f, in part(a)? (c) Can you find a function G(x,y) F(x,) +* ) such that a = , in part (a)? What is ko)? (d) What is ? Can you now find / (x,y)?

Answer 1

$(219, 2)= aproslg3+22 hereo Fryalar2 = 0 sincet is and order contains polynomial in al. if we find 39 yd or more derivay then fzusinal=0 tay contains the polynomial inor fros, y, z) = 12 cos(93 +2²) at here to check fayzzz zo here is we final derivative, it contains the term sin(y3+2²) ft rosly 3+22) So cannot be zero, toryzaz does not vanishes (zero) since it contains costg²+²), sincy3t2) for = 3x + ay2, fy= baytay sult fy Cl, 1) =3 of has continuous mixed second partial derivatives fnyt tye by clairout's theorem, I fony - Eyol i no a) fy (1, 1) = bxX1 +2x1=3 b+2=3 fny = tyre 204 = by jaa=b 2051 a a= 1

Cellpage Unte Page NO: fas= 3n+ 1 y fy- nytay b) consider Fras, y) = 328 toy²e - 372 + 1 yase (i integrate poo se are A G 4oy is a уу? - antay 2 = 3x + 2 y 2 ad22 2 c) cr(2, 4) = 3 212 + 4 here kly) = y fy= nytay ауа +2y = 2y tay = f 17 04 act - 62 -3xt-42 3X+Y a tal then ao - for, du afy - fra, y) = 3 2 2 + 1 y se ty

Summary polynomial is az cosity 3+22), sinly? tay contains the -2 2y2017 fryzzz contains 2y222 40- 3.०) 12 = 30412 y:0y+RY = Fref cht helt & = (held of P Key 22 a -24: 39+Y fcony - ३०२२ CX