Physicsnoob90 said:Homework Statement
Find the local maximum and minimum values and saddle points of the function
f(x,y) = x^2 + xy + y^2 + y
Homework Equations
Local max/min
critical points
saddle
The Attempt at a Solution
1) partial derivative: fx = 2x+y fy = x+2y+1
from here, I'm a little confuse on what should i do to find the critical point in order to solve for the local max/min.
A local maximum value is a point on a graph where the function reaches its highest value in a specific region. This means that within a small neighborhood of the point, the function is higher at that point than at any other nearby points.
To find local minimum values, you can use a similar method to finding local maximum values. A local minimum value is a point on a graph where the function reaches its lowest value in a specific region. This means that within a small neighborhood of the point, the function is lower at that point than at any other nearby points.
A saddle point is a point on a graph where the function has both a local maximum and a local minimum. This means that the function is increasing in some directions and decreasing in others. Visually, a saddle point looks like a saddle on the graph.
To determine the type of critical point, you can use the second derivative test. This involves taking the second derivative of the function and evaluating it at the critical point. If the second derivative is positive, the critical point is a local minimum. If the second derivative is negative, the critical point is a local maximum. If the second derivative is zero, the test is inconclusive and further analysis is needed.
Yes, a function can have multiple local maximum or minimum values. This occurs when the function has multiple peaks or valleys in different regions. It is also possible for a function to have no local maximum or minimum values, or to have only one local maximum or minimum value.