Question: "Find the second partial derivatives for the function..."

College
Calculus and Analysis
Differential Calculus
Derivatives

CoormaBak9

Answered question

2021-04-24

Find the second partial derivatives for the function

f (x, y) = x^{4} - 3 x^{2} y^{2} + y^{2}

and evaluate it at the point(1,0).

Answer & Explanation

Gennenzip

Skilled2021-04-26Added 96 answers

Step 1
Given function is
$f (x, y) = x^{4} - 3 x^{2} y^{2} + y^{2}$
Firstly, we find first order partial derivatives
$f_{x} (x, y) = 4 x^{3} - 6 x y^{2}$
$f_{y} (x, y) = - 6 x^{2} y + 2 y$
Step 2
Now, we find second order partial derivatives
$f_{x x} (1, 0) = 12 {(1)}^{2} - 6 {(0)}^{2} = 12$
$f_{y y} (1, 0) = - 6 {(1)}^{2} + 2 = - 4$
$f_{x y} (1, 0) = - 12 (1) (0) = 0$
$f_{y x} (1, 0) = - 12 (1) (0) = 0$
Step 4
Answer:
$f_{x x} = 12$
$f_{y y} = - 4$
$f_{x y} = 0$
$f_{y x} = 0$

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