Find out the answer to "Find the smallest and largest values and points..."

generals336

Answered question

2021-10-10

Find the smallest and largest values and points of inflection of the function.
$f (x, y) = y^{4} + 4 y^{2} - x^{2}$

Answer & Explanation

casincal

Skilled2021-10-11Added 82 answers

Step 1
The given function is $f (x, y) = y^{4} + 4 y^{2} - x^{2}$
To find the local minimum and maximum values and saddle points of the function first find the partial derivatives.
Step 2
$f_{x} = - 2 x, f_{y} = 4 y^{3} + 8 y$
Now, equate them to 0. That is:
$- 2 x = 0; 4 y^{3} + 8 y = 0$
$\Rightarrow x = 0; 4 y (y^{2} + 2) = 0$
$\Rightarrow x = 0; y = 0 (∵ (y^{2} + 2) i s n e v e r z e r o)$
So, the function has the critical point (0,0).
Now, $f_{x x} = - 2, f_{y y} = 12 x^{2} + 8, f_{x y} = 0$
At(0,0),
$D = (f_{x x}) (f_{y y}) - {(f_{x y})}^{2} = (- 2) (0) - (0)$
=-2
Here, at (0,0) D is negative. Hence, (0,0) is the saddle point. And there is no local maxima and minima.

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