Help with squareroot algebra

[tue]

Hi all

I have two functions and I want to calculate where they intersect. In order to do so, I have taken the derivative of both functions and set 0 = function 1 - funtcion 2. However I cant solve for the derivatives even though is seems quite simple.

I hope someone can help me.

a, b, c and d are constants, and I want to solve for X:

0 = a + b/x² + c + 2dx


Thanks for all suggestions in advance

 

[DrPhil]

Hi all

I have two functions and I want to calculate where they intersect. In order to do so, I have taken the derivative of both functions and set 0 = function 1 - funtcion 2. However I cant solve for the derivatives even though is seems quite simple.

I hope someone can help me.

a, b, c and d are constants, and I want to solve for X:

0 = a + b/x² + c + 2dx


Thanks for all suggestions in advance

You do NOT set the derivatives equal - just set f₁(x) = f₂(x), and solve that for the x of the intersection(s).

 

[tue]

Maybe I explained poorly. But the functions are:

f1(x) = ax + bx^0,5
f2(x) = cx + dx_²

So one function is quadratic and the other with a squareroot. They both have an optimum which I can calculate separetaly, but when the functions are added, I want to calculate the new optimum which is where the derivative of the combined function = 0.

JOined function f(x) = ax + bx^0,5 + cx + dx²

f'(x) = a+0,5bx^-0,5+c +2dx

My overall problem is thus to solve when I have a term with 1/x^0,5


​any ideas?

 

[lookagain]

Maybe I explained poorly. But the functions are:

f1(x) = ax + bx^0,5
f2(x) = cx + dx_²

So one function is quadratic and the other with a squareroot. They both have an optimum which I can calculate separetaly, but when the functions are added, I want to calculate the new optimum which is where the derivative of the combined function = 0.

JOined function f(x) = ax + bx^0,5 + cx + dx²

f'(x) = a+0,5bx^-0,5+c +2dx

My overall problem is thus to solve when I have a term with 1/x^0,5


​any ideas?

a + 0.5b/x^0,5 + c + 2dx = 0

Let y = x^0,5


Then \(\displaystyle y^2 = x.\)


\(\displaystyle a + 0.5b/y + c + 2dy^2 = 0\)


\(\displaystyle 2dy^2 + 0.5b/y + (a + c) = 0 \)


Multiply both sides by 2y:


\(\displaystyle 4dy^3 + b + 2(a + c)y = 0\)


\(\displaystyle 4dy^3 + 2(a + c)y + b = 0\)


You'd have a cubic at his point. If you were to have appropriate coefficients,
you could use the Rational Root Theorem.

 

[tue]

Thanks for the suggestion. Is there no way of avoiding a cube root or a way to solve the cube root?

It should be added that the solution should only be in real numbers, and I know that the coefficients take the following values:

a and d < 0
b and c > 0

 

[stapel]

I have two functions and I want to calculate where they intersect.

Maybe I explained poorly.... I want to calculate the new optimum which is where the derivative of the combined function = 0.

Maybe it would help if you provided the exact text of the exercise...? Because you seem to be wanting to do two entirely different things...? :-?

 

[tue]

Its not an exercise, its a model for a project im working on

The idea is that I have two firms, with each their own way of optimizing profit. This is function 1 and 2.

They share X, which is the amount of credit they give to each other.
The intuition is that if one firm optimize for itself, the other company suffers and vice versa. Therefore they have to work together.

So if both firms are to prosper, they have to look at the combined function:

Combined function f(x) = ax + bx^0,5 + cx + dx²

So simply, the optimal point is where the derivative is = 0, as in any other optimization problem. But the derivative seems more complicated than the original function

f'(x) = a+0,5bx^-0,5+c +2dx

At the moment I calculate the optimum in excel, using their solver function. Simply something like "optimize f(x) by changing the value of x"
This works fine and when I draw the functions it also looks nice.

For example, if the value are:

a=-0,1
b = 0,5
c = 0,05
d = -0,001

Then x = 11,6393 which gives the optimum.

BUt I would like to find this solution using algebra

 

[JeffM]

Its not an exercise, its a model for a project im working on

The idea is that I have two firms, with each their own way of optimizing profit. This is function 1 and 2.

They share X, which is the amount of credit they give to each other.
The intuition is that if one firm optimize for itself, the other company suffers and vice versa. Therefore they have to work together.

So if both firms are to prosper, they have to look at the combined function:

Combined function f(x) = ax + bx^0,5 + cx + dx²

So simply, the optimal point is where the derivative is = 0, as in any other optimization problem. But the derivative seems more complicated than the original function

f'(x) = a+0,5bx^-0,5+c +2dx

At the moment I calculate the optimum in excel, using their solver function. Simply something like "optimize f(x) by changing the value of x"
This works fine and when I draw the functions it also looks nice.

For example, if the value are:

a=-0,1
b = 0,5
c = 0,05
d = -0,001

Then x = 11,6393 which gives the optimum.

BUt I would like to find this solution using algebra

Are you aware that there is a whole economic literature on the behavior of oligopolies generally and duopolies specifically?

Notice that x is constrained: it cannot be less than 0 nor more than e + f, the respective capacities of each firm. So you have left some parameters out, namely e and f. Furthermore, you have not shown that the value of x that optimizes f(x) is an equilibrium value. Indeed, you have not shown that there is an equilibrium value, let alone that it is an attractor. An attracting equilibrium, if it exists, will be a Nash equilibrium, one where neither firm benefits from changing x. The fundamental point is that neither firm has any interest in maximizing the profits of the other. Firm A may prefer a solution that optimizes its profits even though that solution causes losses for Firm B.

The problem that you have devised for yourself is much more complicated than you have imagined. Furthermore, your intuition that the equilibrium solution will maximize the sum of the profits of two independent firms is actually not well founded.

 

[tue]

Are you aware that there is a whole economic literature on the behavior of oligopolies generally and duopolies specifically?

Notice that x is constrained: it cannot be less than 0 nor more than e + f, the respective capacities of each firm. So you have left some parameters out, namely e and f. Furthermore, you have not shown that the value of x that optimizes f(x) is an equilibrium value. Indeed, you have not shown that there is an equilibrium value, let alone that it is an attractor. An attracting equilibrium, if it exists, will be a Nash equilibrium, one where neither firm benefits from changing x. The fundamental point is that neither firm has any interest in maximizing the profits of the other. Firm A may prefer a solution that optimizes its profits even though that solution causes losses for Firm B.

The problem that you have devised for yourself is much more complicated than you have imagined. Furthermore, your intuition that the equilibrium solution will maximize the sum of the profits of two independent firms is actually not well founded.

Thanks a lot for the comments.
This is maybe a bit outside the scope of the original question, but anyways.
The optimization problem is not for production capacity so I havent let e+f out. I have let out constants for both functions, showing the "base profitability" of each company, but that doesnt change the problem.
You are however right that x >= 0, but again this doesnt help me.
The situation is as you mentioned at a nash equilibrium where both firms will have no incentive to change their strategy, but this is what my project is all about, changing the behaviour of both firms for their mutual benefit.

When this is said, the above funtions are not carved in stone, but they need some certain properties which is described below:

The function for the first company. a has to be negative and b has to be positive. I've chosen to use the squareroot as it gives diminishing values.
f1(x) = ax + bx^0,5

The function for the second company. c has to be positive and d has to be negative. I've chosen to use the square as it gives increasing values.
f2(x) = cx + dx_²


So in summary I've chosen the formulaes for their properties and nice symmetry, but If you know of any simpler models which would give the same properties and also preferably symmetry I would be glad to know.

It should also be mentioned that the function should work where x=0, so I cannot replace the squareroot with fx. 1/X

I have attached an image showing how the function should look like. X is on the x-axis. The blue line is function1, the red line is function2 and the green line is the two functions together divided by 2.