# Travis Henderson's Question: Optimizing f(x,y,z) with Constraint

• MHB
• MarkFL
In summary, the maximum value of $f(x,y,z)$ is 1, and the minimum value is $\dfrac{1}{3}$, subject to the constraint $x^2+y^2+z^2=1$.
Hello Travis,

We are given the objective function:

$f(x,y,z)=x^4+y^4+z^4$

subject to the constraint:

$g(x,y,z)=x^2+y^2+z^2-1=0$

Using Lagrange multipliers, we obtain the system:

$4x^3=\lambda(2x)$

$4y^3=\lambda(2y)$

$4z^3=\lambda(2z)$

We see that 12 critical points arise when one of the variables is zero, and the other two are not zero. We see that the other two have to be equal, and their value is found from the constraint:

$y^2+x^2=1$

$x=y=\pm\frac{1}{\sqrt{2}}$

The 12 critical points come from the permutations of:

$\displaystyle \left(0,\pm\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}} \right),\,\left(\pm\frac{1}{\sqrt{2}},0,\pm\frac{1}{\sqrt{2}} \right),\,\left(\pm\frac{1}{\sqrt{2}},\pm\frac{1}{\sqrt{2}},0 \right)$

The objective function's value is the same at each of the 12 points and is given by:

$f_1=\dfrac{1}{2}$

We also see that there are 6 critical values that arise from two of the varaibles being zero, and the other one being $\pm1$. They are:

$(0,0,\pm1),\,(0,\pm1,0),\,(\pm1,0,0)$

The objective function's value is the same at each of the 12 points and is given by:

$f_2=1$

Lastly the other 8 critical values comes from:

$x=y=z$

and substituting into the constraint, we find:

$x=y=z=\pm\dfrac{1}{\sqrt{3}}$

and so we have the 8 permutations of:

$f_3=f\left(\pm\dfrac{1}{\sqrt{3}},\pm\dfrac{1}{ \sqrt{3}},\pm\dfrac{1}{\sqrt{3}} \right)=\dfrac{1}{3}$

Hence we find:

$f_{\text{min}}=\dfrac{1}{3}$

$f_{\text{max}}=1$

Last edited:

## 1. What is the main goal of optimizing f(x,y,z) with constraint?

The main goal of optimizing f(x,y,z) with constraint is to find the maximum or minimum value of the function f(x,y,z) while staying within the given constraints. This allows for the most efficient or effective solution to a problem.

## 2. How is the optimization process for f(x,y,z) different from a regular optimization?

The optimization process for f(x,y,z) with constraint is different because it involves finding the optimal solution while also considering the constraints that must be met. This requires additional steps, such as incorporating the constraints into the objective function or using specialized optimization techniques.

## 3. What are some common constraints in optimizing f(x,y,z)?

Some common constraints in optimizing f(x,y,z) include inequalities (e.g. x ≥ 0), equalities (e.g. x + y + z = 10), and bounds on the variables (e.g. 0 ≤ x ≤ 10). These constraints can also be expressed as mathematical equations or geometric shapes.

## 4. How do you know if the optimized solution for f(x,y,z) is the global optimum?

In most cases, it is not possible to know for certain if the optimized solution for f(x,y,z) is the global optimum. However, using specialized optimization algorithms and techniques can increase the likelihood of finding the global optimum. Additionally, sensitivity analysis can be performed to determine the effect of small changes in the constraints on the optimized solution.

## 5. Can optimization with constraints be applied to real-world problems?

Yes, optimization with constraints is commonly used in real-world problems, such as in engineering, economics, and logistics. It allows for finding the most efficient or effective solution while considering real-world limitations and constraints. Examples include maximizing profits while minimizing costs, or determining the optimal route for a delivery truck while considering time and distance constraints.

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