Question

Solve for x and y in the inequality x^4 + y^4 ≥ 4xy + 2.

a. x = 2, y = 2
b. x = 1, y = 1
c. x = 0, y = 0
d. x = -1, y = -1

Answer 1

To solve the inequality x^4 + y^4 ≥ 4xy + 2, we can factor and simplify the expression. The solution is x = 0, y = 0.

Step-by-step explanation:

To solve the inequality x^4 + y^4 ≥ 4xy + 2:

Subtract 4xy from both sides of the inequality:

x^4 + y^4 - 4xy ≥ 2

Factor the left side of the inequality using the identity a^2 - 2ab + b^2 = (a - b)^2:

(x^2 - y^2)^2 ≥ 2

Take the square root of both sides of the inequality:

x^2 - y^2 ≥ √2

Factor the left side of the inequality using the difference of squares formula a^2 - b^2 = (a + b)(a - b):

(x + y)(x - y) ≥ √2

Since x + y and x - y are both non-negative, the product (x + y)(x - y) will be non-negative:

(x + y)(x - y) ≥ 0

So, for the inequality (x + y)(x - y) ≥ √2 to be satisfied, either both factors must be non-negative, or both factors must be non-positive. Therefore, the answer is c. x = 0, y = 0.