Analytical Applications of Differentiation

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Employing first derivative tests across all intervals helps determine when increasing costs outweigh benefits irrespective of scale economies or diseconomies.

Applying inflection points via concavity changes indicated by second derivative signs provides insight into diminishing returns thresholds.

Resorting solely to average cost functions versus marginal analysis might miss specific outputs where marginal gains diminish.

Utilizing integration to find areas under marginal cost curves until reaching total costs could give optimal production levels despite diminishing returns.

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Constraint equation can be written as $V = (l + x)(w + x)x$ taking into account additional material added when folding.

The constraint equation is given by $V = l w h$ where $h$ represents height after folding without considering cuts.

If squares with side length $x$ are cut out from each corner of paper sized $l$ by $w$, then $V = x(l - 2x)(w - 2x)$.

Volume maximization requires setting up constraints as $V = l w - 4x^2$ indicating area lost by cutting corners only.

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Point directly above or below second point on curve.

Point where first derivative equals zero.

Lowest point of the curve.

Point at intersection with Y axis.

$12\pi$

$32$

$16$

$8\sqrt{2}$

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Setting partial derivatives $\frac{\partial S}{\partial r}$ & $\frac{\partial S}{\partial h}$ equal recognizes dependence between variables during constrained optimization.

Considering separate minimizations for base circumference vs lateral surface disregards fixed relation imposed through constant volume condition.

Analyzing cross-sections through vertical slicing & comparing their circumferences suggests direct proportionality between $r$ & $h$.

Assuming linear relationships between dimensions ignores non-linear nature posed by surface area equations like those derived using pyramids or prisms.