Consider the function e^-alpha(x^2 + y^2), where alpha is a positive constant Li
ID: 1511530 • Letter: C
Question
Consider the function e^-alpha(x^2 + y^2), where alpha is a positive constant Like entropy, this function is positive everywhere but has a maximum for some value of the two variables. For what value of (x, y) is there a maximum? What is the value of partial differential f/partial differential x at this point? What is the value of partial differential f/partial differential y at this point? Is partial differential^2 f/partial differential x^2 at this point positive or negative? Is partial differential^2 f/partial differential y^2 at this point positive or negative?Explanation / Answer
a) For the function to be maximum, (x2 + y2) has to be minimum. So, the function is maximum at (0, 0).
b) f/x = -2xf
At x = 0 and y = 0, f/x = 0
c) f/x = -2yf
At x = 0 and y = 0, f/y = 0
d) 2f/x2 = (-2xf)/x = -2[x(f/x) + f] = -2[x(-2xf) + f] = -2f(1 - 2x2)
At x = 0 and y = 0, 2f/x2 = -2f < 0
e) 2f/y2 = (-2yf)/y = -2[y(f/y) + f] = -2[y(-2yf) + f] = -2f(1 - 2y2)
At x = 0 and y = 0, 2f/y2 = -2f < 0