Step by step solution to "Analyzing critical points. Use the Second Derivative Test..."

vestirme4

Answered question

2021-10-12

Analyzing critical points.
Use the Second Derivative Test to classify the critical points of

f (x, y) = x^{2} + 2 y^{2} - 4 x + 4 y + 6

Answer & Explanation

mhalmantus

Skilled2021-10-13Added 105 answers

Step 1
Given: $f (x, y) = x^{2} + 2 y^{2} - 4 x + 4 y + 6$
To find the critical points : $f_{x} = 0$ and $f_{y} = 0$
$f_{x} = 0 \Rightarrow 2 x - 4 = 0 \Rightarrow x = 2$ and
$f_{y} = 0 \Rightarrow 4 y + 4 = 0 \Rightarrow y = - 1$
Step 2
Analyzing the second derivative test using the critical point: $(x, y) = (2, - 1)$
Let $A = f_{\times}, B = f_{x y},$ and
$C = f_{y y}$
$A = f_{\times} = 2, B = f_{x y} = 0$ and $C = f_{y y} = 4$
$A C - B^{2} = 8 - 0 = 8 > 0$
Therefore, at the point (2, -1) f is local maximum.

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