Find and classify all the critical points of given function f(x,y) f(x,y)= x sin

Question

Find and classify all the critical points of given function f(x,y)
f(x,y)= x sin(x+y)

Explanation / Answer

First, take the first order partial derivatives: f_x = sin(x+y) + x cos(x+y), and f_y = x cos(x+y). For the critical points, set these equal to 0: sin(x+y) + x cos(x+y) = 0, and x cos(x+y) = 0. Hence, sin(x+y) = 0 and x cos(x+y) = 0. (i) If x = 0, then we need sin(0+y) = 0 ==> y = np for any integer n. (ii) Otherwise, we need sin(x+y) = 0 and cos(x+y) = 0, which has no solution. So, the critical points are at (x, y) = (0, np) for any integer n. ------------------ Next, we classify these with the Second Derivative Test. Since f_x = sin(x+y) + x cos(x+y), and f_y = x cos(x+y), we have f_xx = 2 cos(x+y) - x sin(x+y) f_yy = -x sin(x+y) f_xy = cos(x+y) - x sin(x+y) ==> D = f_xx * f_yy - (f_xy)^2 ..........= [2 cos(x+y) - x sin(x+y)][cos(x+y) - x sin(x+y)] - (-x sin(x+y)^2). Note that D(0, np) = 2 cos^2(np) = 2 > 0, and f_xx (0, np) = 2 cos(np). (i) If n is even, then D(0, np) > 0, and f_xx (0, np) = 2 > 0. ==> We have a local minimum. (ii) If n is odd, then D(0, np) > 0, and f_xx (0, np) = -2 < 0. ==> We have a local maximum.

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