Question 664592
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Multiply length times width times height:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \left(24\ -\ 2x\right)^2\left(x\right)\ =\ \left(4x^2\ -\ 96x\ +\ 576\right)\left(x\right)\ =\ 4x^3\ -\ 96x^2\ +\ 576x]


So


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ V(x)\ =\ 4x^3\ -\ 96x^2\ +\ 576x]


Take the first derivative (Power and Sum Rules)


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{dV}{dx}\ =\ 12x^2\ -\ 192x\ +\ 576]


Set the first derivative equal to zero and solve.  This is a quadratic so you will get two roots, each of which represents the abscissa of a local extremum of the original function.


Take the second derivative:


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ \frac{d^2V}{dx^2}\ =\ 24x\ -\ 192]


Evaluate


*[tex \LARGE \ \ \ \ \ \ \ \ \ \ f^{\left[2\right]}(a) ]


where *[tex \LARGE a] is either of the abscissas of the possible extreme points.   If *[tex \LARGE f^{\left[2\right]}(a) \ <\ 0] then *[tex \LARGE a] is the abscissa of a local maximum, if *[tex \LARGE f^{\left[2\right]}(a)\ >\ 0] then *[tex \LARGE a] is the abscissa of a local minimum.  If *[tex \LARGE f^{\left[2\right]}(a)\ =\ 0], the second derivative test is inconclusive, but it is a possible inflection point.   In the latter case, for this problem, calculate the volume for the value of *[tex \LARGE a] in question and see if it makes sense as a possible *[tex \LARGE x] value that could give a maximum volume.


John
*[tex \LARGE e^{i\pi}\ +\ 1\ =\ 0]
My calculator said it, I believe it, that settles it
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