HELP PLEASE Which equation represents a line which is perpendicular to the line y=-2x-3

sergey [27]

Answer:

1) (-3,-2)

2) x=-3

Step-by-step explanation:

1) The vertex of a parabola is the turning point that is the minimum or maximum of the graph, based on the shape of the parabola. In this case, the vertex is the minimum. By looking at the coordinate points, we can tell that the minimum value where the graph changes the direction is at (-3,-2).

2) The line of symmetry is the line on the graph that cuts the shape exactly in half. This means that if you were to fold the graph along this line, the two sides would be identical. All parabolas have a line of symmetry and it always matches the vertex. The equation for the line of symmetry will be x= whatever the x-value of the vertex is. So, for this graph, the line of symmetry is x=-3.

7 0

3 years ago

Find the critical points of the surface f(x, y) = x3 - 6xy + y3 and determine their nature.

Vedmedyk [2.9K]

Compute the gradient of $f$ .

$\nabla f(x,y) = \left\langle 3x^2 - 6y, -6x + 3y^2\right\rangle$

Set this equal to the zero vector and solve for the critical points.

$3x^2-6y = 0 \implies x^2 = 2y$

$-6x+3y^2=0 \implies y^2 = 2x \implies y = \pm\sqrt{2x}$

$\implies x^2 = \pm2\sqrt{2x}$

$\implies x^4 = 8x$

$\implies x^4 - 8x = 0$

$\implies x (x-2) (x^2 + 2x + 4) = 0$

$\implies x = 0 \text{ or } x-2 = 0 \text{ or } x^2 + 2x + 4 = 0$

$\implies x = 0 \text{ or } x = 2 \text{ or } (x+1)^2 + 3 = 0$

The last case has no real solution, so we can ignore it.

Now,

$x=0 \implies 0^2 = 2y \implies y=0$

$x=2 \implies 2^2 = 2y \implies y=2$

so we have two critical points (0, 0) and (2, 2).

Compute the Hessian matrix (i.e. Jacobian of the gradient).

$H(x,y) = \begin{bmatrix} 6x & -6 \\ -6 & 6y \end{bmatrix}$

Check the sign of the determinant of the Hessian at each of the critical points.

$\det H(0,0) = \begin{vmatrix} 0 & -6 \\ -6 & 0 \end{vmatrix} = -36 < 0$

which indicates a saddle point at (0, 0);

$\det H(2,2) = \begin{vmatrix} 12 & -6 \\ -6 & 12 \end{vmatrix} = 108 > 0$

We also have $f_{xx}(2,2) = 12 > 0$ , which together indicate a local minimum at (2, 2).

3 0

2 years ago

$3,254,107 what digit is in the hundred thousands place

Sergio [31]

The digit 2 would be in the hundred thousands place because if you say it out loud its "three million, two hundred fifty four thousand, one hundred and seven"

5 0

3 years ago

Read 2 more answers

How do you slove 3x+4y=8 in steps

gavmur [86]

3x+4y=8
Add -4y to both sides
3x+4y+−4y=8+−4y3x=−4y+8
Then you divide both sides by 3
3x/3=−4y+8/3x=−4/3y+8/3
And your answer is ...
x=−4/3y+8/3

7 0

3 years ago

Alonzo runs 6 miles in 50 minutes. at the same rate, how many miles would he run in 35 minutes?

mihalych1998 [28]

Answer: 4.2 miles

Step-by-step explanation:

6 mi/50 minutes = 6/50 mi per minute

35 minutes * 6/50 = 4.2 miles

5 0

3 years ago