MATH105 April 2017
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Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?

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Solution

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The method of Lagrange multipliers states that a maximum in the interior of the domain must occur at a place where the following equations are satisfied:
${\begin{aligned}{\frac {\partial f}{\partial x}}&=\lambda {\frac {\partial {G}}{\partial {x}}}\\{\frac {\partial f}{\partial y}}&=\lambda {\frac {\partial {G}}{\partial {y}}}\\G(x,y)&=0\end{aligned}}$.
We now compute the derivatives:
${\begin{aligned}{\frac {\partial f}{\partial x}}&=4x^{3/4}y^{3/4}\\{\frac {\partial G}{\partial x}}&=50\\{\frac {\partial f}{\partial y}}&=12x^{1/4}y^{1/4}\\{\frac {\partial G}{\partial y}}&=100\end{aligned}}$.
So we need to solve the system
${\begin{aligned}4x^{3/4}y^{3/4}&=50\lambda \\12x^{1/4}y^{1/4}&=100\lambda \\50x+100y&=500000\end{aligned}}$.
If we multiply the first equation by 2, we get
$8x^{3/4}y^{3/4}=100\lambda$.
Substituting this into the second equation gives
$12x^{1/4}y^{1/4}=8x^{3/4}y^{3/4}$.
Finally, we multiply both sides by $x^{3/4}y^{1/4}$ to arrive at the equation
$12x=8y$,
and solve for y to get
$y={\frac {3}{2}}x.$
We now plug into the third equation:
$50x+100\cdot {\frac {3}{2}}x=500000$.
This simplifies to
$200x=500000$
which comes out to
$x=2500.$
From this, it is clear that
$y=3750$
Giving a value of
$f(x,y)=16(2500)^{1/4}(3750)^{3/4}$.
This can be further simplified to
$f(x,y)=20000\cdot 54^{1/4}.$
According to the problem statement, this is all we need to do  the problem stated that we do not need to provide justification. We will provide a justification here nonetheless. The family of points satisfying G(x,y) = 0 form a line in the plane. The family of such points with $x\geq 0$ and with $y\geq 0$ are a line segment with endpoints on the yaxis and xaxis. Because the maximum must either occur at a critical point or at an endpoint of the domain, and because f(x,y) = 0 if either x or y is zero, it follows that f achieves its maximum at the critical point (2500, 3750).
Answer: $\color {blue}20000{\sqrt[{4}]{54}}$


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