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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:
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