Partial Derivatives Maximum and Minimum Values

In summary, to find the absolute maximum and minimum values of f on the set D, you must first find the critical points of f in D. In this case, both partial derivatives of f are constants and cannot be set equal to 0, so there are no critical points. Therefore, you must proceed to step 2 and find the extreme values of f on the boundary of D. The largest of these values will be the absolute maximum value, and the smallest will be the absolute minimum value.
  • #1
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Homework Statement


Find the absolute maximum and minimum values of f on
the set D.

f(x,y) = 1+4x-5y
D is the closed triangular region with vertices (0,0) (2,0) (0,3)

Homework Equations


To find the absolute maximum and minimum values of a continuous function
on a closed, bounded set :
1. Find the values of f at the critical points of f in D.
2. Find the extreme values of f on the boundary of D.
3. The largest of the values from steps 1 and 2 is the absolute maximum value;
the smallest of these values is the absolute minimum value.

The Attempt at a Solution



So I know to find the values of f at the critical points I must take partial fx and partial fy. Which both should be 0.

But fx(x,y) = 4 and fy(x,y) = -5. How do I set those equal to 0?
 
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  • #2
As you have deduced those equations do not have solutions. That means that there are no critical points of f, so you must continue to step 2.
 

FAQ: Partial Derivatives Maximum and Minimum Values

1. What is a partial derivative?

A partial derivative measures the rate of change of a function with respect to one of its variables while holding all other variables constant. It is denoted by ∂ (the partial symbol) followed by the variable with respect to which the derivative is being taken.

2. How are partial derivatives used to find maximum and minimum values?

In order to find maximum and minimum values of a multivariable function, we take partial derivatives with respect to each variable and set them equal to 0. Then, we solve the resulting system of equations to find the critical points. The maximum or minimum values can then be determined by plugging these critical points back into the original function.

3. What is a critical point?

A critical point is a point on a function where both partial derivatives are equal to 0. This means that the rate of change in all directions is 0, and thus the function may have a maximum, minimum, or saddle point at this location.

4. How do you determine if a critical point is a maximum, minimum, or saddle point?

To determine the nature of a critical point, we can use the second derivative test. This involves taking the second partial derivatives with respect to each variable and evaluating them at the critical point. If the resulting value is positive, the critical point is a minimum. If it is negative, the critical point is a maximum. If it is 0, the nature of the critical point is inconclusive and further analysis is needed.

5. Can a function have multiple maximum or minimum values?

Yes, a function can have multiple maximum and minimum values. This can occur when there are multiple critical points or when the function has a flat region where the partial derivatives are equal to 0, but the second derivative test is inconclusive. In these cases, further analysis is needed to determine the nature of the critical points and the corresponding maximum and minimum values.

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