Maximizing/Minimizing Points of z=-3x^2+54x+52y-3xy-2y^2+100

  • Thread starter MichaelB1301
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In summary, to find the maximum or minimum point of a function, take the derivative, set it equal to 0, and solve for the variable. The purpose of this is to find the highest or lowest point on the graph, which can have real-life applications. The second derivative can determine if a function has a maximum or minimum point, and the coefficient of the squared terms affects the shape of the graph. The maximum or minimum point can only occur at a critical or inflection point, not at the endpoints of the domain.
  • #1
MichaelB1301
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How do you max/min points of the following equation

z=-3x^2+54x+52y-3xy-2y^2+100

regards

mb
 
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  • #2
Find the points where dz/dx = dz/dy = 0. Use the second partial derivative test to check whether they are minima or maxima.
 
  • #3
I would add that this isn't an implicit function. It's just two-variable. It's explicity defined
z=f(x,y) so finding the partials is pretty straight forward.
 

1. How do I find the maximum or minimum point of a function?

To find the maximum or minimum point of a function, you will need to first take the derivative of the function and set it equal to 0. Then, solve for the variable and plug the value back into the original function to find the point.

2. What is the purpose of maximizing or minimizing a function?

The purpose of maximizing or minimizing a function is to find the highest or lowest point on the graph. This can be useful in many real-life applications, such as maximizing profits or minimizing costs.

3. How can I determine if a function has a maximum or minimum point?

A function will have a maximum point if the second derivative is negative and a minimum point if the second derivative is positive. You can also look at the graph of the function to determine if there is a peak or valley.

4. What is the significance of the coefficient of the squared terms in a function?

The coefficient of the squared terms in a function affects the shape of the graph. A positive coefficient will result in a parabola opening upwards, indicating a minimum point, while a negative coefficient will result in a parabola opening downwards, indicating a maximum point.

5. Can the maximum or minimum point of a function occur at the endpoints of the domain?

No, the maximum or minimum point of a function can only occur at a critical point, where the derivative is equal to 0, or at an inflection point, where the second derivative changes sign. Endpoints of the domain cannot be critical points.

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