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dolphi86 [110]
3 years ago
5

Functions 1 and 2 are shown below: Function 1: f(x) = −3x2 + 2 A graph of a parabola with x intercepts of negative 0.5, 0 and 2,

0 and a vertex of 0.5, 4 is shown. Which function has a larger maximum? Type your answer as 1 or 2.
Mathematics
2 answers:
frozen [14]3 years ago
7 0

Answer:

The maximum value of function 1 is 2 and  maximum value of function 2 is 4. So, function 2 has a larger maximum.

Step-by-step explanation:

In Function 1 :

f(x)=-3x^2+2

Lending coefficient is negative. It is a downward parabola and vertex is the maximum point.

vertex of a parabola ax^2+bx+c=0 is

vertex=(\frac{-b}{2a},f(\frac{-b}{2a}))

On comparing with given equation we ave;

a = -3 , b=0 and c = 2

then;

x =\frac{-0}{2(-3)} =0

Substitute the value of x =0 in f(x)=-3x^2+2 we have;

f(0)=-3(0)^2+2=0+2 = 2

⇒Vertex = (0, 2)

⇒the maximum value of function 1 is 2.

In Function 2:

Vertex = (0.5, 4)

⇒the maximum value of function 2 is 4.

Therefore, Function 2 has a larger maximum.

ANEK [815]3 years ago
6 0
2. i belive the answer should be

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The functions q and r are defined as follows.<br> HELP PLEASE
dusya [7]

Answer:

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Step-by-step explanation:

First do r(3) by replacing x with 3 in r:

2*3^2 + 2 = 2*9 + 2 = 18 + 2 = 20

Then replace r(3) with 20 to do q(20):

-20-2 = -22

7 0
3 years ago
The plane x+y+2z=8 intersects the paraboloid z=x2+y2 in an ellipse. Find the points on this ellipse that are nearest to and fart
DiKsa [7]

Answer:

The minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

Step-by-step explanation:

Here, the two constraints are

g (x, y, z) = x + y + 2z − 8  

and  

h (x, y, z) = x ² + y² − z.

Any critical  point that we find during the Lagrange multiplier process will satisfy both of these constraints, so we  actually don’t need to find an explicit equation for the ellipse that is their intersection.

Suppose that (x, y, z) is any point that satisfies both of the constraints (and hence is on the ellipse.)

Then the distance from (x, y, z) to the origin is given by

√((x − 0)² + (y − 0)² + (z − 0)² ).

This expression (and its partial derivatives) would be cumbersome to work with, so we will find the the extrema  of the square of the distance. Thus, our objective function is

f(x, y, z) = x ² + y ² + z ²

and

∇f = (2x, 2y, 2z )

λ∇g = (λ, λ, 2λ)

µ∇h = (2µx, 2µy, −µ)

Thus the system we need to solve for (x, y, z) is

                           2x = λ + 2µx                         (1)

                           2y = λ + 2µy                       (2)

                           2z = 2λ − µ                          (3)

                           x + y + 2z = 8                      (4)

                           x ² + y ² − z = 0                     (5)

Subtracting (2) from (1) and factoring gives

                     2 (x − y) = 2µ (x − y)

so µ = 1  whenever x ≠ y. Substituting µ = 1 into (1) gives us λ = 0 and substituting µ = 1 and λ = 0  into (3) gives us  2z = −1  and thus z = − 1 /2 . Subtituting z = − 1 /2  into (4) and (5) gives us

                            x + y − 9 = 0

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however, x ² + y ² +  1 /2  = 0  has no solution. Thus we must have x = y.

Since we now know x = y, (4) and (5) become

2x + 2z = 8

2x  ² − z = 0

so

z = 4 − x

z = 2x²

Combining these together gives us  2x²  = 4 − x , so

2x²  + x − 4 = 0 which has solutions

x =  (-1+√33)/4

and

x = -(1+√33)/4.

Further substitution yeilds the critical points  

((-1+√33)/4; (-1+√33)/4; (17-√33)/4)   and

(-(1+√33)/4; - (1+√33)/4; (17+√33)/4).

Substituting these into our  objective function gives us

f((-1+√33)/4; (-1+√33)/4; (17-√33)/4) = (195-19√33)/8

f(-(1+√33)/4; - (1+√33)/4; (17+√33)/4) = (195+19√33)/8

Thus minimum distance of   √((195-19√33)/8)  occurs at  ((-1+√33)/4; (-1+√33)/4; (17-√33)/4)  and the maximum distance of  √((195+19√33)/8)  occurs at (-(1+√33)/4; - (1+√33)/4; (17+√33)/4)

4 0
3 years ago
Which value must be added to the expression x2 – 3x to make it a perfect-square trinomial?
Vitek1552 [10]

Answer:

\frac{9}{4}

Step-by-step explanation:

Given

x² - 3x

To make the expression a perfect square

add ( half the coefficient of the x- term )²

x² + 2( - \frac{3}{2} )x + \frac{9}{4}, thus

x² - 3x + \frac{9}{4}

= (x - \frac{3}{2})² ← a perfect square

7 0
3 years ago
An inheritance of $40,000 is divided among three investments yielding $3,500 in interest per year. The interest rates for the th
a_sh-v [17]

Answer:

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Step-by-step explanation:

given,

Investment amount $40,000

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second and third are $3,000 and $5,000 less than the first  

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the investment by each

 x = $ 16,000  , y = $13,000  z= $11,000

4 0
3 years ago
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