Help im confused and ty i can give brainliest and extra pts if u want even if u can only answeer 1

(a)5%)Let fx, y) = x^4 + y^4 - 4xy + 1. and classify each critical point Find all critical points of fx,y) as a local minimum, l

lara31 [8.8K]

$f(x,y)=x^4+y^4-4xy+1$

has critical points wherever the partial derivatives vanish:

$f_x=4x^3-4y=0\implies x^3=y$

$f_y=4y^3-4x=0\implies y^3=x$

Then

$x^3=y\implies x^9=x\implies x(x^8-1)=0\implies x=0\text{ or }x=\pm1$

If $x=0$ , then $y=0$ ; critical point at (0, 0)
If $x=1$ , then $y=1$ ; critical point at (1, 1)
If $x=-1$ , then $y=-1$ ; critical point at (-1, -1)

$f(x,y)$ has Hessian matrix

$H(x,y)=\begin{bmatrix}12x^2&-4\\-4&12y^2\end{bmatrix}$

with determinant

$\det H(x,y)=144x^2y^2-16$

At (0, 0), the Hessian determinant is -16, which indicates a saddle point.
At (1, 1), the determinant is 128, and $f_{xx}(1,1)=12$ , which indicates a local minimum.
At (-1, -1), the determinant is again 128, and $f_{xx}(-1,-1)=12$ , which indicates another local minimum.

4 0

3 years ago

Click an item in the list or group of pictures at the bottom of the problem and, holding the button down, drag it into the corre

Natasha_Volkova [10]

xy - -4x²

Step-by-step explanation:

(2x - 3y)(4x -y)

2x( 4x -y) -3x( 4x - y)

8x² - 2xy - 12x² + 3xy

8x²- 12x²- 2xy + 3xy

-4x² + xy

xy - -4x²

6 0

3 years ago

Solve for x:<br> x + 3x - 19 = 17

Natasha_Volkova [10]

<h2><u>Answer:</u></h2><h2><u></u> $\boxed{\boxed{\sf{x=9}}}$ <u></u></h2><h2><u></u></h2><h2><u>Solution Steps:</u></h2>

______________________________

<h3>1.) Combine x and 3x:</h3>

$x+3x=4x$

<em> - We combine the </em> $(x)$ <em> variables because to get to the end sum of the equation, there can only be 1 variable. So to get the 1, you combine like terms. </em>

<em />

<h3>2.) Add 19 to both sides:</h3>

$-19+19=$ Cancels Out

<em></em> $17+19=36$ <em></em>

<em> - We do this to get 1 number left over. To do this you do the opposite of what you see. In this case we have a negative 19, so you add 19 on each side of the equals. </em>

<em />

<h3>3.) Divide both sides by 4:</h3>

$4x$ ÷ $4=x$

$36$ ÷ $4=9$

<em> - We do this to get x alone. To do this you do the opposite of what you see. In this case we have </em> $4(x)$ <em> which is multiplying, so you divide by 4 on each side. </em>

<em />

So your answer is $\bold{x=9}$ .

______________________________

7 0

3 years ago

12.) Last week on

Yuri [45]

Answer:

75 feet long

Step-by-step explanation:

3 0

3 years ago

Please help asap 25 pts

nikitadnepr [17]

$4y-8-2y+5=0\\\\(4y-2y)+(-8+5)+0\\\\2y-3=0\qquad|\text{add 3 to both sides}\\\\2y=3\qquad|\text{divide both sides by 2}\\\\y=\dfrac{3}{2}\\\\\boxed{y=1\dfrac{1}{2}}\to\boxed{c.}$

4 0

3 years ago

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