I am so tilted! Please help me! (Multivariate Calculus about finding Extrema!)

Question

Good day!

Show that f(x,y) = 1 - x^4 - y^4 has a relative maximum, even though D(Discriminant) in the theorem is 0.

JUST HOW!!!!!! I AM TILTED AF PLEASE HELP ME!

I will show my working so far:

fx = -4x^3 here x = 0

fy = -4y^3 here y = 0

fxx = -12x^2 since x = 0, the value of fxx would be 0 as well.

fyy = -12y^2

fxy = 0

D = fxx x fyy - fxy^2 = 0 (Which should mean that not enough information is given for this function.)

AND THIS STUPID QUESTION TELLS ME TO SHOW THAT THERE IS RELATIVE MAXIMUM!

Answer 1

Your work so far is correct. Unfortunately, the second derivative test is inconclusive when D = 0. In this case, we can show that there is a global maximum for the function.

First, we observe that x⁴ ≥ 0 for all x and y⁴ ≥ 0 for all y. Therefore, (x⁴ + y⁴) ≥ 0 for all x,y.

Next we manipulate our function to show that it has a maximum:

f(x,y) = 1 - x⁴ - y⁴

= 1 - (x⁴ + y⁴) (factor -1 out of the 2nd and 3rd terms)

f(x, y) + (x⁴ + y⁴) = 1 (add (x⁴ + y⁴) to both sides)

(x⁴ + y⁴) = 1 - f(x,y) (subtract f(x,y) from both sides)

1 - f(x,y) ≥ 0 (substitute for (x⁴ + y⁴) in the inequality (x⁴ + y⁴) ≥ 0)

1 ≥ f(x,y) (add f(x,y) to both sides)

f(x,y) ≤ 1

Now that we have shown a global maximum, we can show that (0,0) is a relative maximum.

f(0,0) = 1

f(x,y) ≤ f(0,0)

Therefore, (0,0) is a relative maximum.