Answer:
Step-by-step explanation:
Given that,
g(x, y) = 5 - (x - 3)² - (y + 2)²
Let find the grad of the function,
The grad of a function is defined as
∇g= ∂g/∂x •i + ∂g/∂y •j + ∂g/∂z •k
∇g = gx•i + gy•j + gz•k
gx = -2(x-3) = -2x+6
gy = -2(y+2) = -2y -4
∇g = (-2x+6) •i + (-2y-4)•j
We have a maximum or a minimum If g conservative, then, ∇g = 0i +0j
Then comparing this to the grad of the function
(-2x+6) •i + (-2y-4)•j = 0i +0j
Then, -2x+6 = 0
2x=6
x = 3
Also, -2y-4=0
-2y=4
y = -2
Then, g(x, y) = 5 - (x - 3)² - (y + 2)²
g(x, y) = 5 - (3 - 3)² - (-2 + 2)²
g(x, y) = 5
So the critical point is (3, -2, 5)
gx =-2x+6
Second derivative of gx with respect to x
gxx=-2
gy=-2y-4
Second derivative of gy with respect to y
gyy=-2
Second derivative of gx with respect to y
gxy =0
d =gxxgyy - (gxy)²
Then, gyy=gxx
d = -2×-2 -0²
d = 4-0=4
Since d>0
Since d is greater than 0, then, it is not a chair points.
Then, since gxx=-2<0, gyy=-2<0
Cause the second derivative in x (or in y) is less than zero, then the point is relatively maximum
So the maximum point is (3, -2, 5).
