Find the critical points of the following functions. f(x,\ y)=x^{4}+y^{4}-4x-32y+10

Alyce Wilkinson

Alyce Wilkinson

Answered question

2021-10-14

Find the critical points of the following functions.
f(x, y)=x4+y44x32y+10

Answer & Explanation

likvau

likvau

Skilled2021-10-15Added 75 answers

Step 1
The function given is f(x, y)=x4+y44x32y+10
Take the first derivative of the function,
fx(x, y)=4x34
4x34=0
4(x31)=0
x=1
fy(x, y)=4y332
4y332=0
4(y38)=0
y38=0
y=2
Step 2
The critical are: (1, 2)
Take the second derivative of the fx(x, y)
f×(x, y)=12x2
At (1, 2),
f×(1, 2)=1212
=12
Take the second derivative of the fy(x, y)
fyy(x, y)=12y2
Step 3
At (1, 2)
fyy(1, 2)=1222
=48
The other second derivative with respect to y,
fxy(x, y)=0
Consider the condition,
D=f×fyy(fxy)2
=12480
=576
Here, the value of discriminant is greater than 0. Therefore the critical point (1, 2) is a local minimum point.

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