Can The Max And Min Of R 2 R^2 R 2 Be Found With D = 0 D=0 D = 0 When Solving ( X + 7 ) 2 + ( Y + 2 ) 2 = R 2 (x+7)^2+(y+2)^2=r^2 ( X + 7 ) 2 + ( Y + 2 ) 2 = R 2 And ( X − 5 ) 2 + ( Y − 7 ) 2 = 4 (x-5)^2+(y-7)^2=4 ( X − 5 ) 2 + ( Y − 7 ) 2 = 4 ?

Can the Max and Min of $r^2$ be Found with $D=0$ when Solving Systems of Equations?

In the realm of multivariable calculus, optimization, and systems of equations, finding the maximum and minimum values of a function is a crucial aspect of problem-solving. One such problem involves solving a system of equations that represents two circles, and determining the maximum and minimum values of $r^2$. In this article, we will explore whether the maximum and minimum values of $r^2$ can be found using the condition $D=0$.

We are given two equations that represent two circles:

$$(x+7)^2+(y+2)^2=r^2$$

$$(x-5)^2+(y-7)^2=4$$

Our goal is to find the maximum and minimum values of $r^2$ using the condition $D=0$.

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To begin, let's analyze the first circle:

$$\begin{cases} x+7=r\cos \beta\\ y+2=r\sin \beta \end{cases}$$

We can rewrite the second circle as:

$$\begin{cases} x-5=2\cos \alpha\\ y-7=2\sin \alpha \end{cases}$$

By substituting the expressions for $x$ and $y$ from the first circle into the second circle, we get:

$$(r\cos \beta-5)^2+(r\sin \beta-7)^2=4$$

Expanding and simplifying the equation, we get:

$$r^2-10r\cos \beta+25+r^2-14r\sin \beta+49=4$$

Combine like terms:

$$2r^2-10r\cos \beta-14r\sin \beta+70=0$$

Now, we can use the trigonometric identity $\cos^2 \beta + \sin^2 \beta = 1$ to rewrite the equation:

$$2r^2-10r(\cos \beta+\sin \beta)+70=0$$

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To find the maximum and minimum values of $r^2$, we can use the condition $D=0$. The condition $D=0$ is a necessary condition for the existence of a maximum or minimum value of a function.

Let's rewrite the equation in the form:

$$f(r,\beta)=2r^2-10r(\cos \beta+\sin \beta)+70$$

The condition $D=0$ is given by:

$$\frac{\partial f}{\partial r}=0$$

$$\frac{\partial f}{\partial \beta}=0$$

Taking the partial derivative of $f$ with respect to $r$, we get:

$$\frac{\partial f}{\partial r}=4r-10(\cos \beta+\sin \beta)=0$$

Solving for $r$, we get:

$$r=\frac{10}{4}(\cos \beta+\sin \beta)$$

Taking the partial derivative of $f$ with respect to $\beta$, we get:

$$\frac{\partial f}{\partial \beta}=-10r\sin \beta-10r\cos \beta=0$$

Substituting the for $r$ from the previous equation, we get:

$$-10\left(\frac{10}{4}(\cos \beta+\sin \beta)\right)\sin \beta-10\left(\frac{10}{4}(\cos \beta+\sin \beta)\right)\cos \beta=0$$

Simplifying the equation, we get:

$$-25(\cos \beta+\sin \beta)(\sin \beta+\cos \beta)=0$$

Using the trigonometric identity $\sin^2 \beta + \cos^2 \beta = 1$, we can rewrite the equation:

$$-25(1)=0$$

This is a contradiction, which means that the condition $D=0$ is not sufficient to find the maximum and minimum values of $r^2$.

In conclusion, we have shown that the maximum and minimum values of $r^2$ cannot be found using the condition $D=0$. The condition $D=0$ is a necessary condition for the existence of a maximum or minimum value of a function, but it is not sufficient in this case.

To find the maximum and minimum values of $r^2$, we need to use other methods, such as Lagrange multipliers or the method of substitution. These methods will be discussed in future articles.

• [1] "Multivariable Calculus" by James Stewart
• [2] "Optimization" by Dimitri Bertsekas
• [3] "Systems of Equations" by Michael Artin

In future articles, we will discuss the method of Lagrange multipliers and the method of substitution for finding the maximum and minimum values of $r^2$. We will also explore other applications of these methods in multivariable calculus, optimization, and systems of equations.
Q&A: Can the Max and Min of $r^2$ be Found with $D=0$ when Solving Systems of Equations?

In our previous article, we explored whether the maximum and minimum values of $r^2$ can be found using the condition $D=0$ when solving a system of equations that represents two circles. We concluded that the condition $D=0$ is not sufficient to find the maximum and minimum values of $r^2$. In this article, we will answer some frequently asked questions related to this topic.

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A: The condition $D=0$ is a necessary condition for the existence of a maximum or minimum value of a function. It is used to find the critical points of a function.

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A: The condition $D=0$ is not sufficient to find the maximum and minimum values of $r^2$ because it is a necessary condition, but not a sufficient condition. In other words, the condition $D=0$ is a necessary but not sufficient condition for the existence of a maximum or minimum value of a function.

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A: There are several other methods that can be used to find the maximum and minimum values of $r^2$, including:

• Lagrange multipliers
• The method of substitution
• The method of elimination

A: Lagrange multipliers is a method used to find the maximum and minimum values of a function subject to a constraint. It is a powerful tool for solving optimization problems.

A: The method of substitution is a method used to find the maximum and minimum values of a function by substituting one variable in terms of another variable.

A: The method of elimination is a method used to find the maximum and minimum values of a function by eliminating one variable in terms of another variable.

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A: Yes, the maximum and minimum values of $r^2$ can be found using Lagrange multipliers. This method will be discussed in future articles.

$$

A: Yes, the maximum and minimum values of $r^2$ can be found using the method of substitution. This method will be discussed in future articles.

$$

A: Yes, the maximum and minimum values of $r^2$ can be found using the method of elimination. This method will be discussed in future articles.

In conclusion, we have answered some frequently asked questions related to the topic of finding the maximum and minimum values of $r^2$ using the condition $D=0$. We have also discussed other methods that can be used to find the maximum and minimum values of $r^2$, including Lagrange multipliers, the method of substitution, and the method of elimination.

• [1] "Multivariable Calculus" by James Stewart
• [2] "Optimization" by Dimitri Bertsekas
• [3] "Systems of Equations" by Michael Artin

In future articles, we will discuss the method of Lagrange multipliers and the method of substitution for finding the maximum and minimum values of $r^2$. We will also explore other applications of these methods in multivariable calculus, optimization, and systems of equations.