Mathematics, 03.07.2019 09:30 janeliles

Find the local maximum and minimum values and saddle point(s) of the function. if you have three-dimensional graphing software, graph the function with a domain and viewpoint that reveal all the important aspects of the function. (enter your answers as a comma-separated list. if an answer does not exist, enter dne.) f(x, y) = 9 - 2x + 4y - x2 - 4y2

Answers: 1

Show answers

Answers

Answer from: ipadguy1806

Solution:

There is no saddle point (DNE). However, there is local maximum at (1, 1/2) for the given function.

Explanation:

we have function of two variables f(x,y)= 9-2x+4y-x^2-4y^2

we will find the values by partial derivative with respect to x,y,xy

$f_{x}$ = -2 -2x

$f_{y}$ = 4 -8y

to find the saddle point we should first find the critical points so equate

-2 -2x=0 and 4 -8y=0

we get x= 1 and y =1/2 so, critical points are (1,1/2)

to find local maximum or minimum we have to find $f_{xx}$ , $f_{yy}$ and $f_{xy}$

formula is $f_{xx}$ * $f_{yy}$ - $f^{2_{xy} }$ =0

$f_{xx}$ = -2

$f_{yy}$ = -8

$f^{2_{xy} }$ =0

putting values in formula

(-2)*(-8) -0 =16 > 0, and $f_{xx}$ < 0 and $f_{yy}$ <0

so, here we have local maximum

we have no saddle point for this function by using the same formula we used to find extrema.

Find the local maximum and minimum values and saddle point(s) of the function. if you have three-dim

Answer from: qmsadler01

The point (-1,1/2) is a local maximum and there are no local minimums and saddle points.

Step-by-step explanation:

The function to study is f(x,y) = 9-2x+4y-x^2-4y^2 which is a differentiable function on the variables and . We find the possible points of local maximum, local minimum and saddle points finding the point where the partial derivatives are zero at the same time. So, we must calculate the partial derivatives:

$\frac{\partial f}{\partial x} = -2-2x = -2(x+1)$

$\frac{\partial f}{\partial y} = 4-8y = 4(1-2y)$

It is not difficult to see that the only point where $\frac{\partial f}{\partial x}(x,y)=\frac{\partial f}{\partial y}(x,y) =0$ is (-1,1/2). Notice that the system of equations

$\begin{cases} -2(x+1) &= 0\\ 4(1-2y) &= 0\end{cases}$

has solution x= -1 and y=1/2.

If we want to know if (-1,1/2) is local maximum, local minimum or saddle point we must calculate the partial derivatives of second order:

$\frac{\partial^2 f}{\partial x^2} = -2 = A$

$\frac{\partial^2 f}{\partial y^2} = -8 = B$

$\frac{\partial^2 f}{\partial y \partial x} = \frac{\partial^2 f}{\partial x \partial y} = 0 = C$ .

Substituting the values into the formula AB-C^2 we get (-2)(-8)-0=16>0. As the value we have obtained is greater than zero and A=-2<0 we assure that the point (-1,1/2) is a local minimum.

There are no local maximums or saddle points.

Answer from: Quest

If it is my birthday, then it is snowing

Answer from: Quest

triangle abc is reflected on the line x=-6.5 then reflected again on the x axis

step-by-step explanation:

Another question on Mathematics

Mathematics, 21.06.2019 15:20

Find the image of a(4, -2) after it is reflected over the line y= 2, then reflected over the line x = 2. (-8,6) o (0.-2) o (0,6) (-8, -2)

Answers: 1

Answer

Mathematics, 21.06.2019 17:00

What expression is equivalent to (2x-1)(x-3)?

Answers: 1

Answer

Mathematics, 21.06.2019 17:00

The following graph shows the consumer price index (cpi) for a fictional country from 1970 to 1980? a.) 1976 - 1978b.) 1972 - 1974c.) 1974 - 1976d.) 1978 - 1980

Answers: 1

Answer

Mathematics, 21.06.2019 19:00

The graph shows the charges for a health club in a month. make a function table for the input-output values. writes an equation that can be used to find the total charge y for the number of x classes

Answers: 2

Answer

You know the right answer?

Find the local maximum and minimum values and saddle point(s) of the function. if you have three-dim...

Questions

Mathematics, 04.03.2021 23:20

The length of the drawn from the vertex of the right angle of the right triangle to its hypotenuse is the geometric mean between th masures of the t...