Solve Lagrange Multipliers Problem w/ e^(9x) - Find (a,b)

In summary, the problem deals with finding the minimum value for a function subject to a constraint, and the solution is found by finding the point on the graph of the function that minimizes the function's gradient.
  • #1
ajkess1994
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I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers:

Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible.

I have found the gradient for both when it's set up as [e^(9x) - y = 0], but now I don't know what to do from here. Would someone please be available to help me?
 
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  • #2
ajkess1994 said:
I have been stumped on this problem and I am probably overthinking it as usual. The problem deals with the Lagrange Multipliers:

Use Lagrange multipliers to find the point (a,b) on the graph of y=e^(9x), where the value (a,b) is as small as possible.

I have found the gradient for both when it's set up as [e^(9x) - y = 0], but now I don't know what to do from here. Would someone please be available to help me?

The objective function is "the value (a,b)" which I interpret to mean the distance from the origin to a point on the curve, and we can use the square of this distance for our objective:

\(\displaystyle f(x,y)=x^2+y^2\)

Subject to the constraint:

\(\displaystyle g(x,y)=e^{9x}-y=0\)

Using Lagrange, we obtain the system:

\(\displaystyle 2x=\lambda\left(9e^{9x}\right)\)

\(\displaystyle 2y=\lambda(-1)\)

This system implies:

\(\displaystyle \frac{2x}{9e^{9x}}=-2y\)

Or:

\(\displaystyle y=-\frac{x}{9e^{9x}}\)

Now, plugging this into the constraint yields:

\(\displaystyle e^{9x}+\frac{x}{9e^{9x}}=0\)

Or:

\(\displaystyle 9e^{18x}+x=0\)

Using a numeric root finding method, we find:

\(\displaystyle x\approx-0.20902833541170100144\)

And so:

\(\displaystyle y\approx0.15239872243985548342\)

We then compute:

\(\displaystyle f(-0.20902833541170100144,0.15239872243985548342)=0.06691821560628668613644254416134077697\)

To ensure we have a minimum, we can use another point on the constraint, such as \((0,1)\), and we find:

\(\displaystyle f(0,1)=1\)

Thus, we may state:

\(\displaystyle f_{\min}\approx0.06691821560628668613644254416134077697\)

Here is a diagram of the solution:

View attachment 8574
 

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  • #3
Thank you for helping me with this problem, enjoy the rest of Veteran's Day
 

FAQ: Solve Lagrange Multipliers Problem w/ e^(9x) - Find (a,b)

What is a Lagrange multiplier?

A Lagrange multiplier is a mathematical tool used to find the maximum or minimum value of a function subject to a set of constraints. It involves finding the critical points of the function and the constraints, and then using a multiplier to account for the constraints in the optimization process.

How do you solve a Lagrange multipliers problem?

To solve a Lagrange multipliers problem, you must first identify the objective function and the constraints. Then, you can set up the Lagrangian function by multiplying each constraint by a Lagrange multiplier and adding it to the objective function. Next, you can find the critical points of the Lagrangian function by taking partial derivatives and setting them equal to zero. Finally, you can use these critical points to solve for the values of the Lagrange multiplier and the variables in the objective function.

What is e^(9x)?

e^(9x) is an exponential function where e is the base and 9x is the exponent. It represents the exponential growth or decay of a quantity over time, where x is the independent variable.

How do you find the critical points of a function?

To find the critical points of a function, you must take the derivative of the function and set it equal to zero. Then, you can solve for the values of the independent variable that make the derivative equal to zero. These values are the critical points of the function.

What is the purpose of using Lagrange multipliers with e^(9x)?

The purpose of using Lagrange multipliers with e^(9x) is to optimize the function while also satisfying the given constraints. This allows for a more efficient and accurate solution to the problem, as the constraints are taken into account during the optimization process.

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