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Vera_Pavlovna [14]
3 years ago
15

Find the absolute maximum and absolute minimum of the function f(x,y)=2x2−4x+y2−4y+1 on the closed triangular plate bounded by t

he lines x=0,y=2,y=2xin the first quadrant.
Mathematics
1 answer:
marusya05 [52]3 years ago
6 0

First check for the critical points of <em>f</em> by checking where the first-order derivatives vanish.

\dfrac{\partial f}{\partial x}=4x-4=0\implies x=1

\dfrac{\partial f}{\partial y}=2y-4=0\implies y=2

Notice how the point (1, 2) lies on the line <em>y</em> = 2<em>x</em> ; at this point, we get a value of <em>f</em>(1, 2) = -5 (MIN).

Next, check the points where the boundary lines intersect, which occurs at the points (0, 0), (0, 2), and (1, 2). We already checked the last one. We find <em>f</em>(0, 0) = 1 (MAX) and <em>f</em>(0, 2) = -3.

Now check on the boundary lines themselves. If <em>x</em> = 0, then

f(0,y)=y^2-4y+1=(y-2)^2-3

which has a maximum value of -3 when <em>y</em> = 2 (so we get the same critical point as before).

If <em>y</em> = 2, then

f(x, 2)=2x^2-4x-3=2(x-1)^2-5

with a maximum of -5 when <em>x</em> = 1.

If <em>y</em> = 2<em>x</em>, then

f(x,2x)=6x^2-12x+1=6(x-1)^2-5

with the same maximum of -5 when <em>x</em> = 1.

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3 years ago
A construction company uses the function f(p) where p is the number of people working on a project to model the amount of money
Jlenok [28]

Answer:

Domain of f(p) =  [0,∞), where it belongs to whole numbers only

Step-by-step explanation:

The domain is the set of all possible values of independent variable for which function is defined

As in the given function f(p), we have the independent variable p. As p is the number of people working on the project, so it means either the number of people could be 0 or it could be anything greater than 0,  like it could be equal to thousand or ten thousand, but it can not be fraction in any case.

So, the domain is set of whole numbers starting from 0.

Domain of f(p) = [0,∞)  


5 0
3 years ago
Read 2 more answers
Solve. <br> 1/2x + 3 &lt; 4x - 7<br><br> A) x &gt; 8/7<br> B) x &lt; -8/7<br> C) x &gt; 20/7
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C would be your answer and here is how you would solve the problem.

5 0
2 years ago
Which of the following best completes the proof showing that ΔWXZ ~ ΔXYZ?
skelet666 [1.2K]

Angles formed by the segment \overline{XZ} in the triangles ΔWXZ, and ΔXYZ, are equal and the given corresponding sides are proportional.

  • The option that best completes the proof showing that ΔWXZ ~ ΔXYZ is; <u>16 over 12 equals 12 over 9</u>

Reasons:

The proof showing that ΔWXZ ~ ΔXYZ is presented as follows;

Segment \overline{XZ} is perpendicular to segment \overline{WY}

∠WZX and ∠XZY are right angles by definition of \overline{XZ}  perpendicular to \overline{WY}

∠WZX in ΔWXZ = ∠XZY in ΔXYZ = 90° (definition)

\displaystyle \frac{WZ}{XZ} = \frac{16}{12} = \mathbf{ \frac{4}{3}}

\displaystyle \mathbf{ \frac{XZ}{ZY}}  = \frac{12}{9}  = \frac{4}{3}

Therefore;

  • \displaystyle \frac{16}{12} = \frac{12}{9}, which gives, \displaystyle \mathbf{\frac{WZ}{XZ} }= \frac{XZ}{ZY}

Given that two sides of ΔWXZ are proportional to two sides of ΔXYZ, and

that the included angles between the two sides, ∠WZX and ∠XZY are

congruent, the two triangles, ΔWXZ and ΔXYZ are similar by Side-Angle-

Side, SAS, similarity postulate.

The option that best completes the proof is therefore;

  • \displaystyle \frac{16}{12} = \frac{12}{9} which is; <u>16 over 12 equals 12 over 9</u>

Learn more about the SAS similarity postulate here:

brainly.com/question/11923416

4 0
2 years ago
How many different arrangements can be made with ten pieces of silverware laid in a row if three are identical spoons, four are
LuckyWell [14K]

10 pieces of silverware laid in a row if 3 are identical spoons, 4 are identical forks, and 3 are identical knifes

The arrangement of 'm' objects on which 'n' objects are of same kind is \frac{m!}{n!}

Given: 10 pieces of silverware so its 10!

3 are identical spoons so its 3!

4 are identical forks so its 4!

and 3 are identical knifes so its 3!

Arrangements made = \frac{10!}{3!*4!*3!} = 4200




4 0
3 years ago
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