Question: Given \(z=100 x^{1 / 4} y^{3 / 4}\), where \(x=48-4 y\),

1. use the chain rule to find \(\frac{d z}{d y}\). Leave your answer in terms of \(x\) and \(y\), Solve \(\frac{d z}{d y}=0\) for \(y\) in terms of \(x\), and use the constraint equation to find \(x\) and \(y\) to maximize z.
2. Calculate both \(\left.\frac{\partial z}{\partial y}\right|_{(12,9)}\) and \(\left.\frac{d z}{d y}\right|_{(12,9)} \quad\) (i.e. at \(\left.x=12, y-9\right)\)

Then complete the following statements interpreting your results:

1. If \(x=48-4 y\), then the rate of change of \(z\) w.r.t. \(y\) at \((x, y)=(12,9)\) is
2. \(\mathrm{At}(x, y)=(12,9)\), if \(x\) is held constant at \(x=12\), then the rate of change of \(z\) w.r.t. \(y\) is

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