Quadratic equation to find max and min

In summary: Okay, I'll just provide the summary for now.In summary, the problem is to find the maximum and minimum values of ##f(x, y) = 4x + y^2## subject to the constraint ##2x^2 + y^2 = 4##. There are a few different ways to approach this problem, such as solving for one variable in the constraint and then substituting it into the function, or using Lagrange multipliers. However, there appears to be some confusion with the solutions, as x = 4 and y = undefined are incorrect. It would be beneficial to sketch the ellipse and understand the domain before attempting to solve the problem.
  • #36
Pi-is-3 said:
Also, if you ever need, ## - \sqrt{a^2+b^2} \leq acos(\theta)+bsin(\theta) \leq \sqrt{a^2+b^2}##
How about this?
 
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  • #37
Lifeforbetter said:
How about this?

Do you mean the proof? You can prove it through Lagrange multipliers, or their is a trigonometrical proof too. For the Lagrange multipliers proof, your optimization function is ##acos(\theta)+bsin(\theta)## and restrain function is ##cos^2(\theta)+sin^2(\theta)=1##.
 
<h2>1. What is a quadratic equation?</h2><p>A quadratic equation is a mathematical expression of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is used to solve problems involving finding the maximum or minimum value of a quadratic function.</p><h2>2. How do you find the maximum or minimum value of a quadratic equation?</h2><p>To find the maximum or minimum value of a quadratic equation, you can use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively. This value of x will give you the x-coordinate of the vertex, which is the maximum or minimum point on the curve.</p><h2>3. What is the significance of the maximum and minimum values in a quadratic equation?</h2><p>The maximum or minimum values in a quadratic equation represent the highest or lowest point on the curve, respectively. This point is also known as the vertex and is important in determining the behavior and shape of the quadratic function.</p><h2>4. Can a quadratic equation have more than one maximum or minimum value?</h2><p>No, a quadratic equation can have only one maximum or minimum value. This is because the graph of a quadratic function is a parabola, which has a single vertex. However, it is possible for the maximum or minimum value to occur at multiple points on the x-axis.</p><h2>5. How is the quadratic equation used in real-world applications?</h2><p>The quadratic equation is used in various fields, such as engineering, physics, and economics, to solve problems involving finding the maximum or minimum value of a function. It can be used to optimize the production of goods, determine the trajectory of a projectile, or find the maximum profit for a business.</p>

1. What is a quadratic equation?

A quadratic equation is a mathematical expression of the form ax^2 + bx + c = 0, where a, b, and c are constants and x is the variable. It is used to solve problems involving finding the maximum or minimum value of a quadratic function.

2. How do you find the maximum or minimum value of a quadratic equation?

To find the maximum or minimum value of a quadratic equation, you can use the formula x = -b/2a, where a and b are the coefficients of the x^2 and x terms, respectively. This value of x will give you the x-coordinate of the vertex, which is the maximum or minimum point on the curve.

3. What is the significance of the maximum and minimum values in a quadratic equation?

The maximum or minimum values in a quadratic equation represent the highest or lowest point on the curve, respectively. This point is also known as the vertex and is important in determining the behavior and shape of the quadratic function.

4. Can a quadratic equation have more than one maximum or minimum value?

No, a quadratic equation can have only one maximum or minimum value. This is because the graph of a quadratic function is a parabola, which has a single vertex. However, it is possible for the maximum or minimum value to occur at multiple points on the x-axis.

5. How is the quadratic equation used in real-world applications?

The quadratic equation is used in various fields, such as engineering, physics, and economics, to solve problems involving finding the maximum or minimum value of a function. It can be used to optimize the production of goods, determine the trajectory of a projectile, or find the maximum profit for a business.

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