Question

Consider the function f(x,y)=y²−2y+x2²+C(xy−x), where C is some constant. (a) Show that if C=±2, then f has only one critical point. (b) Classify for which value of C=±2 this critical point is a maximum, a minimum or a saddle point. (c) Set C=0. Find the max and minimum value of the function in the boundary of the semicircle D:={(x,y)∣y=2−x2​}. (d) Set C=0. Find the max and minimum value of the function in semicircle D:={(x,y)∣y=2−x2​}

Answer 1

To determine the critical points and their nature for the function f(x,y), we consider the partial derivatives and the second derivative test, particularly for C=±2. For C=0, we evaluate the function on and within the boundary of semicircle D to find extremal values.

Step-by-step explanation:

Identifying Critical Points and Classifying Extrema

For the function f(x,y)=y²-2y+x^4+C(xy-x), with C as a constant, we need to find the critical points by setting the derivatives of the function with respect to x and y to zero.

For (a), if C=±2, the partial derivatives are simplified, and we can solve the system to find that there is only one critical point, which occurs when both partial derivatives are zero.

For (b), we determine the nature of the critical point using the second derivative test. We assess the determinant of the Hessian matrix (the matrix of second-order partial derivatives) at the critical point. The sign of this determinant for C=±2 distinguishes whether we have a maximum, a minimum, or a saddle point.

For (c) and (d), with C=0, we evaluate the function f(x,y) on the boundary of the semicircle D:={(x,y)∣y=2-√x2} and within D to find the maximum and minimum values.