Question

Find the directions in which the function increases and decreases most rapidly at Po. Then find the derivatives of the function in those directions u f(x,y,z) = 4 In (xy) + 2 In (yz)+2 In (xz). Po(1,1,1) In which direction does the function increase most rapidly? (i+1+O (Type exact answers, using radicals as needed) In which direction does the function decrease most rapidly? = (1+) ;+ OK (Type exact answers, using radicals as needed.) What is the value of the derivative in the direction of u? (0) = (Type an exact answer, using radicals as needed.) What is the value of the derivative in the direction of -u?

Answer 1

sop Given function is (Q) iz 4 + 2 4 ta + X 3 ス fluly, 2) = 41n(xy) +21h (yz) taln (x2), Poll, li.) The arreet on which function increases most rapidly U = of=<fnity ifz) 2+2 > У y = 1 §, & Point Poll, 1, 1) U= Foll, 1) = 26 4 >={6,6,4) uz fitoj tuk The direction which function decreases most rapidly is -4 =-bi-61-42 4 Af 6 (6)

Mow have to find value u て (Dut Po in the direction of 4. Now (Duflo & Of(1,1,1). U Since of Chil) = (60664) Now vt 1977 so lofl=562462 py2 = J 36+36+16 Jss Now ů = 6 188 188 Joss 188 Now (Dufllo =<6,6,4). (6 J80 36 +36 + 16 = 88 188 y s 588 . 88 188 (Dut po = 18 (Dut Po=888

C we will Now (Douf) po find the value of the direction of u. (D-uf) Poe wl. (-4 Where A -u a - U tul = 188 - ا --6 Now 1-41 56624 (6) 27 (-4)2 J36436416 LUE-6 -4) J88 788 J88 Now (Duf) Po=(-6 -6 J88 J88 188 J7-16 = 36 + 36 + ho 588 588 -4 1-6, -18-4 8E 188 (D-utpo = JPE