why is (0, a/b) and (c/b, 0) NOT a critical point??
Since (w/(s-w), r/s) does not satisfy the equations for a critical point, how did you get that?rad0786 said:I have been working with the equations:
x' = rx - sxy/(1+tx)
y' = sxy/(1+tx) - wy
I find that the only singular point is (0,0)
is this correct
I got another point, ( w/(s-w) , r/s ), however, that cannot be a critical point because when substituting, x' and y' fail to be 0.
HallsofIvy said:Since (w/(s-w), r/s) does not satisfy the equations for a critical point, how did you get that?
HallsofIvy said:The equations for a critical point are x'= rx- sxy/(1+ tx)= 0 and y'= sxy/(1+ tx)- wy= 0. The first gives sxy/(1+ tx)= rx and the second sxy/(1+ tx)= wy. Since the left sides of both equations are the same, rx= wy. Every point on the line y= (r/w)x (or, if w= 0, the line x= 0) is a singular point.
Critical points in a function refer to the points where the function's derivative is equal to zero or does not exist. These points are important in determining the behavior and shape of a function.
To find the critical points of a function, you need to first take the derivative of the function with respect to the variable in question. Then, set the derivative equal to zero and solve for the variable to find the critical points.
Critical points are directly related to extrema, which refer to the maximum or minimum values of a function. In fact, critical points are often used to help find the location of extrema in a function.
In order to determine if a critical point is a maximum or minimum, you can use the second derivative test. If the second derivative is positive at the critical point, then it is a minimum. If the second derivative is negative, then it is a maximum.
Yes, a function can have multiple critical points. In fact, a polynomial function of degree n can have up to n-1 critical points. It is important to consider all critical points when analyzing the behavior of a function.