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28-12-2010, 08:57 PM | #1 |
+Thành Viên+ Tham gia ngày: Nov 2010 Bài gởi: 280 Thanks: 152 Thanked 77 Times in 49 Posts | Maximum value. Find Max. value of $(a-x)(b-y)(c-z)(ax+by+cz) $. where $a,b,c $ are positive and $(a-x),(b-y) $ and $(c-z) $ are also positive. |
29-12-2010, 12:02 PM | #2 | |
+Thành Viên+ | Trích:
It's obvious that the maximum value of P cannot be satisfying $P_{max} \le 0 $ because P is always greater than 0 where we take any $x,y,z $ such that $0<x<a,0<y<b,0<z<c. $. By the given condition, $(a-x), (b-y) $ and $(c-z) $ are positive. Then $(ax+by+cz) $ is also positive. Consequently, just applying AM-GM inequality for 4 positive real numbers system $(a-x, b-y, c-z, ax + by + cz) $ as will be expressing as below, then make both 2 sides to 4th power, we obtain: $(a - x)(b - y)(c - z)(ax + by + cz) = \frac{1}{{abc}}({a^2} - ax)({b^2} - by)({c^2} - cz)(ax + by + cz) \le \frac{1}{{256abc}}{({a^2} + {b^2} + {c^2})^4} $. The equality holds if and only if there exists some $(x,y,z) $ satisfies the linear combination: $a - x = b - y = c - z = ax + by + cz $. That system exactly has the only solution because after implying to the linear equations system $\left\{ \begin{array}{l}(a + 1)x + by + cz = a \\x - y = a - b \\ y - z = b - c \\\end{array} \right. $, the determinant $D=a+b+c+1>0 $. Thus, it's not very difficult to compute the appropriate $(x,y,z) $. Finally, u should check that whether it would be suitable to the given hypothesis or not. Best regards! Poincare. thay đổi nội dung bởi: Poincare, 29-12-2010 lúc 12:07 PM | |
The Following User Says Thank You to Poincare For This Useful Post: | man111 (29-12-2010) |
29-12-2010, 12:15 PM | #3 | |
+Thành Viên+ | Trích:
But I think the system should be $a^2-ax=b^2-by=c^2-cz=ax+by+cz $ | |
The Following User Says Thank You to leviethai For This Useful Post: | man111 (29-12-2010) |
29-12-2010, 12:24 PM | #4 | |
+Thành Viên+ | Trích:
Thanks for your recommendation! Poincare, | |
The Following User Says Thank You to Poincare For This Useful Post: | man111 (29-12-2010) |
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