chwala said:
chwala said:
Thanks from the plot we have the point ##(-7.540, -5.595)## being an inflection point or can we say saddle point? then ##(0.471, -0.396)## being the local minimum... bringing me to the next question, do we have a global maximum and global minimum for this problem?fresh_42 said:
To find the local maxima and minima of a function, we first calculate the partial derivatives with respect to each variable (in this case, x and y). Then, we set these partial derivatives equal to zero and solve the resulting system of equations to find critical points. Finally, we use the second partial derivative test or another appropriate method to determine whether each critical point is a local maximum, local minimum, or saddle point.
The function f(x, y) = x^3-xy-x+xy^3-y^4 is a multivariable function that depends on the variables x and y. It is a polynomial function that includes terms involving x, y, and both x and y raised to various powers.
To calculate the partial derivatives of the function f(x, y) = x^3-xy-x+xy^3-y^4, we differentiate each term of the function with respect to the corresponding variable (x or y) while treating the other variable as a constant. This process yields the partial derivatives of the function with respect to x and y.
Critical points are points in the domain of a function where the partial derivatives are equal to zero or do not exist. These points are potential candidates for local maxima, local minima, or saddle points of the function. By analyzing the behavior of the function around these critical points, we can determine whether they correspond to maxima, minima, or saddle points.
The second partial derivative test involves calculating the second partial derivatives of a function at a critical point and examining the signs of these second derivatives to determine the nature of the critical point. If the second partial derivatives satisfy certain conditions, we can conclude whether the critical point is a local maximum, local minimum, or saddle point of the function. This test provides a systematic way to classify critical points in terms of their behavior as maxima, minima, or saddle points.