Differentiation Practice Problems - [Optimization] | Online Math Tutoring LLC

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Answer the following questions in regards to determining the optimized solution.

Determine the point on the curve \(y=2-x^2\) that closest to the point \((3,2)\).
What is the area of the largest rectangle whose base resides on the x-axis and the other two vertices on the parabola \(y=6-x^2\).
Determine the maximum value of the product \(xy\) if \(y=\frac{1}{\sqrt{x}}-\sqrt{x}\).

An isosceles triangle whose base is the line segment from \((0,0)\) to \((k,0)\), where \(k>0\), has it vertex on the graph of \(f(x)=\sqrt{16-x^2}\). Determine the value(s) of \(k\) does the triangle have a maximum area. [See Figure]

\((1,1)\)

\(8\sqrt(2)\)

\(\frac{2\sqrt{3}}{9}\)

\(k=2\)