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balu736 [363]
3 years ago
11

yuri wrote twice as many poems as xenia. yuri wrote 3 more poems than xenia. the number of poems written can be represented by t

he system of equations: y=2x and y=x+3
Mathematics
1 answer:
balu736 [363]3 years ago
5 0

Answer:

y = 2x and y = x + 3

2x = x + 3

x = 3

y = 2*3 = 6 or y = 3 + 3 = 6

yuri wrote 6 and xenia wrote 3 poems


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Find the absolute maximum and absolute minimum values of the function f(x, y) = x 2 + y 2 − x 2 y + 7 on the set d = {(x, y) : |
dsp73

Looks like f(x,y)=x^2+y^2-x^2y+7.

f_x=2x-2xy=0\implies2x(1-y)=0\implies x=0\text{ or }y=1

f_y=2y-x^2=0\implies2y=x^2

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The Hessian matrix for this function is

H(x,y)=\begin{bmatrix}f_{xx}&f_{xy}\\f_{yx}&f_{yy}\end{bmatrix}=\begin{bmatrix}2-2y&-2x\\-2x&2\end{bmatrix}

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Now consider each boundary:

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f(1,y)=8-y+y^2=\left(y-\dfrac12\right)^2+\dfrac{31}4

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f(-1,y)=8-y+y^2

and we get the same extrema as in the previous case: 8 at (-1, 1), and 10 at (-1, -1).

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f(x,1)=8

which doesn't tell us about anything we don't already know (namely that 8 is an extreme value).

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which has 3 extreme values, but the previous cases already include them.

Hence f(x,y) has absolute maxima of 10 at the points (1, -1) and (-1, -1) and an absolute minimum of 0 at (0, 0).

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