Implicit Function Homework: Finding Min/Max

In summary, the question is about finding the extrema points and determining if they are minimum or maximum for an implicit function. The first part required finding all the extrema points and the second part involves determining if they are minimum or maximum. The poster has provided the derivative and points found, with the additional condition of F(a) = b. The question is how to find the second derivative of an implicit function and use it to determine if the points are minimum or maximum. The suggested solution is to use the second derivative test.
  • #1
asi123
258
0

Homework Statement



Hey.
I have this problem with the second part of the question.
In the first part, I needed to find all the extrema points of this implicit function and the second one is to find if it's minimum or maximum.
I posted only the derivative of the implicit function and the points I found.
BTW there was another condition to find the point, they said F(a) = b, so together with the derivative, I found the points.
Back to the question, as I was saying, the second part is to find if it's minimum or maximum, how do I do that?
I thought about finding the second derivative and plugging the points into it, but how do you find the second derivative of an implicit function, same as you find the first one?

I didn't posted the question, cause it's kind hard to translate, so I hope I gave you all the info you need.


Homework Equations





The Attempt at a Solution

 

Attachments

  • scan0012.jpg
    scan0012.jpg
    12.3 KB · Views: 346
Physics news on Phys.org
  • #2
Use the second derivative test: If y"(x)>0 x is a minimum, if y"(x)< 0 x is a maximum.
 

1. What is an implicit function?

An implicit function is a mathematical relationship between two variables where one variable is not explicitly written in terms of the other. In other words, the equation cannot be solved for one variable in terms of the other.

2. How do you find the minimum and maximum values of an implicit function?

To find the minimum and maximum values of an implicit function, you can use the following steps:

  • Take the partial derivatives of the function with respect to each variable
  • Set both derivatives equal to zero and solve for the variables
  • Substitute the values of the variables into the original function to find the corresponding minimum or maximum value
  • Check the second partial derivatives to confirm if the values found are minimum or maximum points

3. What is the difference between a local and global minimum/maximum?

A local minimum/maximum is a point on a function where the value is lower/higher than all other points in a small neighborhood. A global minimum/maximum, on the other hand, is the lowest/highest value of the entire function.

4. Can implicit functions have multiple maximum or minimum points?

Yes, it is possible for an implicit function to have multiple maximum or minimum points. This occurs when the second partial derivatives are both positive or both negative at different points on the function.

5. How can implicit functions be applied in real life?

Implicit functions can be used to model relationships between variables that are not directly measurable, such as the relationship between income and spending habits. They can also be used in optimization problems to find the most efficient solution, such as in production processes or resource allocation. Additionally, implicit functions are commonly used in physics and engineering to describe the behavior of systems.

Similar threads

Applying the implicit function theorem to a system of equations
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Lagrange multipliers understanding
    • Calculus and Beyond Homework Help
Replies
8
Views
466
Gradient & Smooth Surfaces: Implicit Function Theorem
    • Calculus and Beyond Homework Help
Replies
16
Views
2K
Checking nature of turning point of parametric equation
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
What is the Solution to Implicit Differentiation Homework with Given Values?
    • Calculus and Beyond Homework Help
Replies
10
Views
1K
Write the tangent lines to the curve
    • Calculus and Beyond Homework Help
Replies
1
Views
882
Finding x/y-intercepts, asymptotes, derivatives, and max min
    • Calculus and Beyond Homework Help
Replies
4
Views
1K
Question about approximate numerical integration methods
    • Calculus and Beyond Homework Help
Replies
4
Views
692
Derivative of a Point of Max: Finding a & b Conditions
    • Calculus and Beyond Homework Help
Replies
7
Views
1K
[Multivariable Calculus] Implicit Function Theorem
    • Calculus and Beyond Homework Help
Replies
2
Views
1K

Hot Threads

Recent Insights

Back
Top