Welcome to Westonci.ca, the Q&A platform where your questions are met with detailed answers from experienced experts. Get immediate and reliable solutions to your questions from a knowledgeable community of professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
Sagot :
From the constraint, we have
[tex]x+y+z=9 \implies z = 9-x-y[/tex]
so that [tex]s[/tex] depends only on [tex]x,y[/tex].
[tex]s = g(x,y) = xy + y(9-x-y) + x(9-x-y) = 9y - y^2 + 9x - x^2 - xy[/tex]
Find the critical points of [tex]g[/tex].
[tex]\dfrac{\partial g}{\partial x} = 9 - 2x - y = 0 \implies 2x + y = 9[/tex]
[tex]\dfrac{\partial g}{\partial y} = 9 - 2y - x = 0[/tex]
Using the given constraint again, we have the condition
[tex]x+y+z = 2x+y \implies x=z[/tex]
so that
[tex]x = 9 - x - y \implies y = 9 - 2x[/tex]
and [tex]s[/tex] depends only on [tex]x[/tex].
[tex]s = h(x) = 9(9-2x) - (9-2x)^2 + 9x - x^2 - x(9-2x) = 18x - 3x^2[/tex]
Find the critical points of [tex]h[/tex].
[tex]\dfrac{dh}{dx} = 18 - 6x = 0 \implies x=3[/tex]
It follows that [tex]y = 9-2\cdot3 = 3[/tex] and [tex]z=3[/tex], so the only critical point of [tex]s[/tex] is at (3, 3, 3).
Differentiate [tex]h[/tex] again and check the sign of the second derivative at the critical point.
[tex]\dfrac{d^2h}{dx^2} = -6 < 0[/tex]
for all [tex]x[/tex], which indicates a maximum.
We find that
[tex]\max\left\{xy+yz+xz \mid x+y+z=9\right\} = \boxed{27} \text{ at } (x,y,z) = (3,3,3)[/tex]
The second derivative at the critical point exists
[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex] for all x, which suggests a maximum.
How to find the maximum value?
Given, the constraint, we have
x + y + z = 9
⇒ z = 9 - x - y
Let s depend only on x, y.
s = g(x, y)
= xy + y(9 - x - y) + x(9 - x - y)
= 9y - y² + 9x - x² - xy
To estimate the critical points of g.
[tex]${data-answer}amp;\frac{\partial g}{\partial x}[/tex] = 9 - 2x - y = 0
[tex]${data-answer}amp;\frac{\partial g}{\partial y}[/tex] = 9 - 2y - x = 0
Utilizing the given constraint again,
x + y + z = 2x + y
⇒ x = z
x = 9 - x - y
⇒ y = 9 - 2x, and s depends only on x.
s = h(x) = 9(9 - 2x) - (9 - 2x)² + 9x - x² - x(9 - 2x) = 18x - 3x²
To estimate the critical points of h.
[tex]$\frac{d h}{d x}=18-6 x=0[/tex]
⇒ x = 3
It pursues that y = 9 - 2 [tex]*[/tex] 3 = 3 and z = 3, so the only critical point of s exists at (3, 3, 3).
Differentiate h again and review the sign of the second derivative at the critical point.
[tex]$\frac{d^{2} h}{d x^{2}}=-6 < 0[/tex]
for all x, which suggests a maximum.
To learn more about constraint refer to:
https://brainly.com/question/24279865
#SPJ9
We hope you found what you were looking for. Feel free to revisit us for more answers and updated information. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
