Analytical Applications of Differentiation

Assuming there exists a cubic polynomial with roots at ${-1,0,p}$ that achieves its absolute maximum over real numbers at $(q,r)$ where $q > p > r > -1$, determine $p$ if $r=\frac{-27}{p}$.

A cylindrical can is to be made to hold 500 cm³ of liquid. What should be the radius of the can to minimize the amount of material used for its construction?

A farmer wants to build a rectangular pen against a straight river, using the river as one side of the pen. If the farmer has 200 meters of fencing and wants to maximize the enclosed area, what should be the dimensions of the rectangular pen?

A cylindrical can is to be made to hold 500 cm³ of liquid. What should be the height of the can to minimize the amount of material used for its construction?

What is the maximum area of a rectangle inscribed in a semicircle of radius 4 if one side lies on the diameter?

A manufacturer wants to produce cylindrical containers with a fixed volume of 1000 cm³. What should be the radius of the container to minimize the amount of material used for its construction?

If you are maximizing the volume of an open box created by cutting squares from each corner of a rectangular sheet of paper and folding up sides, how would you express the constraint equation correctly?

For minimizing surface area $S$ while maintaining constant volume $V$ for cylindrical containers without tops which relationship between radius $r$ & height $h$ needs consideration?

If the function $f(x) = \frac{1}{3}x^3 - cx^2 + 6x$ has a local maximum at $x = 3$, what is the value of $c$?

If a box without a top is to have a volume of $500 ext{ cm}^3$, which expression represents its surface area in terms of its height $(h)$ if its base dimensions are equal?