Question

14. Find three positive numbers x, y, and z such that the sum of the three numbers satisfies the equation 3x+y+2z = 24 and whose product is a maximum.

Answer 1

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Answer – given, sum of 3 numbers satisfy 3x + y + 2z = 248 z= (24 - 3x - y) 2 let product of three numbers is given by f(x,y) i.e. 1 1 f(x, y) = xy (3124 (24-3x - - »)} = }(2479 – 3x+y=xv) (1) we know, # Suppose that (a, b) is a critical point of f(x, y) and that the second order partial derivatives are continuous in some region that contains (a,b) Next de fine, D = D(a,b) = fxx(a,b)fyy(a,b) – [fxy(a,b)]? We then have the following classifications of the critical point. (1) If D > 0 and fxx(a,b) > 0 then there is a relative minimum at (a,b) (2) If D > 0 and fxx(a,b) < 0 then there is a relative maximum at (a,b) (3)If D <0 then the point (a,b) is a saddle point. d Basic Rules -(1) -t" = n. t dt d d (2) c f(t) = c f(t),c = constant dt dt 11-1 a (S дx 2 1 - (2) (3) similarly, fx(x, y) (24xy – 3x²y – xy2) (24 n) – 3y2 () - vare = } [24y(1) – 3y(2x)= y*(1)] $=(x, y) = }(244 – Gry –y?) = faz(x, ) = 24, 0) – 6.2.c 262) = 10–6y(1) – 0] = – 34 = fy(,y) = 2x [] (24xy – 3x²y = xy) a – 3x(y) – x ду [24r(1) – 3x2(1) – x(2y)] = fy(x, y) = }[243 – 3x2 – 2xy] = fyy(x, y) = 5124 f(x) – 3.5+ (x2) – 2x Jy (9) – 0 – 2x] = - * now, f xy(x, y) = 1 (24.0) - co - (-3) - 3241) – 64(1)=(20) => f xy(x, y) = 12 – 3x – y to find critical point, we solve f z(x, y) = 0 & fy(x, y) = 0 i. e. [24y – 6xy – y2] = 0 & -[24x – 3x2 – 2xy] = 0 a 24x (y) ду = (4) a a a = (5) ду (6) 1 8 solving these eqn. we get points (8,0), (0,0), (0, 24) & from eqn(1), D = (-3y)(-x) – (12 – 3x – y) = 3xy + 3x + y - 12 Case(a)for (0,0) -D <0 so can't take this point. Case(b)for (8,0) →D>0 & fxx = 0 so can't say anything. Case(c)for (0,24) →D>0 & fxx < 0 so this will give maximum value. Case(d) for ,8] →D > 0& fxx < 0 so this will give maximum value. Now, at (0,24), f(x, y) = = 0 (240)(24) (24(0)(24) – 3(0)2(24) – (0)(24)2) = (65) (8.5) - }(24()s) -=($) * = (30)697) >0 & at so we will take this pair of number then 8 24-34--8 = 4 *3 =Z= PAS hence, number are W100 i.e. x= 8 y = 8 &z = 4 3' #Ans..