A geologist in Tibet discovers a new mountain in the area of Gyangtse. The equation of the mountain is given by f(x,y) = -x^4 - y^4 +2xy where x and

y is measured in mile. What is the
relative maximum height of the mountain in meters?

you need

Fx = 0 and Fy=0 for a max, so
-4x^3 + 2y = 0
-4y^3 + 2x = 0

max f=1/2 at x = y = ±1/√2

(1137.967, 1137.967) meters

To find the relative maximum height of the mountain, we need to find the maximum value of the function f(x, y) = -x^4 - y^4 + 2xy.

To do this, we can find the critical points of the function, and then determine whether each critical point is a relative maximum or minimum by the second derivative test.

First, let's find the partial derivatives of f with respect to x and y:

∂f/∂x = -4x^3 + 2y
∂f/∂y = -4y^3 + 2x

Next, let's find the critical points by setting both partial derivatives equal to zero:

-4x^3 + 2y = 0
-4y^3 + 2x = 0

Simplifying these equations, we have:
2y = 4x^3
2x = 4y^3

Dividing each equation by 2, we get:
y = 2x^3
x = 2y^3

Now, we can solve these equations simultaneously to find the critical points:

Substituting y = 2x^3 into x = 2y^3:
x = 2(2x^3)^3
x = 2(8x^9)
x = 16x^9
1 = 16x^8
x^8 = 1/16
x = (1/16)^(1/8)
x = 0.5

Substituting x = 0.5 into y = 2x^3:
y = 2(0.5)^3
y = 2(0.125)
y = 0.25

So the critical point is (x, y) = (0.5, 0.25).

Now, let's find the second partial derivatives:

∂^2f/∂x^2 = -12x^2
∂^2f/∂y^2 = -12y^2
∂^2f/∂x∂y = 2

Now we can evaluate the second partial derivatives at the critical point (0.5, 0.25):

∂^2f/∂x^2 = -12(0.5)^2 = -3
∂^2f/∂y^2 = -12(0.25)^2 = -0.75
∂^2f/∂x∂y = 2

The determinant of the Hessian matrix (second derivative matrix) is given by:

D = (∂^2f/∂x^2)(∂^2f/∂y^2) - (∂^2f/∂x∂y)^2
= (-3)(-0.75) - (2)^2
= 2.25 - 4
= -1.75

Since the determinant D is negative, this implies that the critical point (0.5, 0.25) is a saddle point rather than a relative maximum or minimum.

Therefore, there is no relative maximum height of the mountain in meters because the function does not have a maximum point in the given domain.