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anemone
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Find the maximum of $a+b$, given $a^2-1+b^2-3b=0$.
MarkFL said:My solution:
Using Lagrange multipliers, we obtain:
\(\displaystyle 1=\lambda(2a)\)
\(\displaystyle 1=\lambda(2b-3)\)
This implies:
\(\displaystyle a=b-\frac{3}{2}\)
Substituting into the constraint, we obtain:
\(\displaystyle \left(b-\frac{3}{2} \right)^2-1+b^2-3b=0\)
\(\displaystyle 2b^2-6b+\frac{5}{4}=0\)
\(\displaystyle 8b^2-24b+5=0\)
\(\displaystyle b=\frac{6\pm\sqrt{26}}{4}\)
Hence:
\(\displaystyle a=\frac{\pm\sqrt{26}}{4}\)
And so the maximum of the objective function $f(a,b)=a+b$ is:
\(\displaystyle f_{\max}=f\left(\frac{\sqrt{26}}{4},\frac{6+\sqrt{26}}{4} \right)=\frac{3+\sqrt{26}}{2}\)
The quadratic constraint is a mathematical equation that restricts the values of the variables in a problem to be in the form of a quadratic expression. This means that the variables can be raised to the power of 2, but cannot have any higher powers or be multiplied together.
To maximize $a+b$ given a quadratic constraint, you need to use a mathematical technique called Lagrange multipliers. This involves setting up a system of equations using the quadratic constraint and the objective function, and then solving for the optimal values of the variables.
One limitation is that the quadratic constraint can be quite restrictive, as it only allows for solutions in the form of a quadratic expression. This may not always be the most efficient or practical solution for a problem. Additionally, the process of using Lagrange multipliers can be complicated and time-consuming.
Sure, let's say we have the quadratic constraint $x^2+y^2=25$ and the objective function $a+b$. Using Lagrange multipliers, we can set up the system of equations: $a+b-\lambda(x^2+y^2-25)=0$ and $2x=\lambda 2x$ and $2y=\lambda 2y$. Solving this system, we get the optimal values of $a$ and $b$ to be 5 and 20, respectively.
Yes, there are many real-world applications for this problem, especially in the field of economics. For example, it can be used to determine the optimal production levels for a company given certain production costs and demand constraints. It can also be applied in engineering to optimize design parameters for maximum efficiency.