- #1
Albert1
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$x,y,z\in R$
$x+y+z=5---(1)$
$xy+yz+zx=3---(2)$
find $\max(z)$
$x+y+z=5---(1)$
$xy+yz+zx=3---(2)$
find $\max(z)$
thanks , your answer is correct :)jacks said:My solution:
$\bf{x+y+z=5 \Rightarrow x+y = 5-z...(1)}$$\bf{xy+yz+xz=3}$$\bf{(x+y+z)^2=x^2+y^2+z^2+2(xy+yz+xz)=25 \Rightarrow x^2+y^2+z^2=19}$So $\bf{x^2+y^2 = 19-z^2...(2)}$Using Cauchy - Schwartz Inequality$\bf{\left(x^2+y^2\right).\left(1^2+1^2\right)\geq \left(x+y \right)^2}$$\bf{\left(19 - z^2 \right).2 \geq \left( 5-z \right)^2}$$\bf{3z^2-10z-13 \leq 0}$$\displaystyle \bf{3.\left(z - \frac{13}{3} \right).\left(z+1\right)\leq 0}$$\displaystyle \bf{ -1 \leq z \leq \frac{13}{3}}$
So $\displaystyle \bf{Max.(z) = \frac{13}{3}}$
The optimal value of $z$ can be determined by using the simplex method, which involves iteratively solving equations (1) and (2) using linear programming techniques. The goal is to find the maximum value of $z$ that satisfies all the constraints in the system.
The goal of maximizing $z$ is to find the best possible solution for a given system of equations. By maximizing $z$, we are finding the optimal value that satisfies all the constraints and gives the best possible outcome for the problem at hand.
Yes, there are other methods that can be used to solve equations (1) and (2), such as the dual simplex method and the interior-point method. However, the simplex method is the most commonly used method for linear programming problems.
The limitations of maximizing $z$ depend on the constraints and variables in the system. In some cases, the constraints may be too complex or the number of variables may be too large for the simplex method to find an optimal solution. Additionally, the solution may not always be feasible or unique.
The results of maximizing $z$ can be interpreted as the optimal value for the given system. This means that by using the values of the variables that maximize $z$, we can achieve the best possible outcome for the problem at hand while still satisfying all the constraints. Additionally, the values of the variables can provide insights into the relationship between the different components of the system.