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Calculus and Analysis
Differential Calculus
Analyzing functions

Ayaana Buck

Answered question

2021-02-02

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

tafzijdeq

Skilled2021-02-03Added 92 answers

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

Jeffrey Jordon

Expert2021-11-07Added 2605 answers

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