Anonymous From: - Posts: - Votes: - |
There are three positive numbers ,the sum of them is 45,please maximize or minimize the value of x2+y2+3z2 . it is from the book calculus 6th edition by james stewart from chapter 15.7 or i think chapter 15.6. please explain and solve it step by step ,otherwise i will not understand it . |

Anonymous From: - Posts: - Votes: - |
Let x,y and z be the three positive numbers. Then, x + y + z = 45 So z = 45 - (x + y) = 45 - x - y Putting value of z in x2 + y2 + 3z2 = x2 + y2 + 3 (45 - x - y)2 Opening whole square = x2 + y2 + 3 ( 2025 + x2 + y2 - 2(45)(x) + 2(x)(y) - 2(45)(y) ) Simplifying expression = x2 + y2 + 3 ( 2025 + x2 + y2 - 90x + 2xy - 90y ) Openng parenthesis by multiplyign with 3 = x2 + y2 + 6075 + 3x2 +3y2 - 270x + 6xy - 270y (1) Partial differentiate w.r.t x fx = 2x + 6x - 270 + 6y = 8x - 270 + 6y (2) Partial differentiate w.r.t y fy = 2y + 6y + 6x - 270 = 8y + 6x - 270 (3) Put fx = 0 and fy = 0 8x - 270 + 6y = 0 => 8x + 6y = 270 (4) 8y + 6x - 270 = 0 => 6x + 8y = 270 (5) Solving (4) and (5) silmultaneously, 8x + 6y = 270 (x6) => 48x + 36y = 1620 (6) 6x + 8y = 270 (x8) => 48x + 64y = 2160 (7) --------------------------- Subtracting (6) and (7) - 28 y = - 540 => y = - 540/-28 = 19 (approx) Putting value of y in (4) 8x + 6(19)= 180 So, x = 20 Fnding second partial derivatives fxx (x, y) = 8 fxx (20, 19) = 8 (> 0) fyy (x, y) = 8 fyy (20, 19) = 8 fxy (x, y) = 6 fxy (20, 19) = 6 Applying the Second Partial Derivative Test D = fxx (x, y)fxy (x, y) - {fxy (x, y) } 2 D = (8)(8) - (6)2 = 64 - 36 = 28 (> 0) x = 20, y = 19 and z = 45 - x - y = 6 So we see that for (20, 19, 6) the function has a relative minimum. |

ProfessorMonkey From: US Posts: 3 Votes: 3 |
Step1 Let f(x,y,z) = x2+y2+3z2 +λ(x+y+z-45); Put first order partial derivatives with respect to x,y,z equal to 0. Step2 2x + λ=0 ; x=-λ/2 ; 2y+λ=0 ; y=-λ/2 ; 6z+λ=0 ; z=-λ/6 Step3 Put these values of x,y,z in x+y+z=45 Step4 -λ/2 -λ/2 -λ/6 =45 ; λ=-270/7 Step5 x= 135/7 ; y= 135/7 and z= 45/7 Step6 fxx =2 ; fyy=2 ; fzz=6 ; A minima exists. |

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