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?

If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!

Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
 If you are stuck on a problem: Read the solution slowly and as soon as you feel you could finish the problem on your own, hide it and work on the problem. Come back later to the solution if you are stuck or if you want to check your work.
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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: $\color {blue}20000{\sqrt[{4}]{54}}$
