A box with a square base and open top must have a volume of 42592 cm. We wish to find the dimensions of the box that minimize the amount of material used.
Minimization
In mathematics, traditional optimization problems are typically expressed in terms of minimization. When we talk about minimizing or maximizing a function, we refer to the maximum and minimum possible values of that function. This can be expressed in terms of global or local range. The definition of minimization in the thesaurus is the process of reducing something to a small amount, value, or position. Minimization (noun) is an instance of belittling or disparagement.
Maxima and Minima
The extreme points of a function are the maximum and the minimum points of the function. A maximum is attained when the function takes the maximum value and a minimum is attained when the function takes the minimum value.
Derivatives
A derivative means a change. Geometrically it can be represented as a line with some steepness. Imagine climbing a mountain which is very steep and 500 meters high. Is it easier to climb? Definitely not! Suppose walking on the road for 500 meters. Which one would be easier? Walking on the road would be much easier than climbing a mountain.
Concavity
In calculus, concavity is a descriptor of mathematics that tells about the shape of the graph. It is the parameter that helps to estimate the maximum and minimum value of any of the functions and the concave nature using the graphical method. We use the first derivative test and second derivative test to understand the concave behavior of the function.
![A box with a square base and open top must have a volume of 42592 cm³. We wish to find the
dimensions of the box that minimize the amount of material used.
First, find a formula for the surface area of the box in terms of only a, the length of one side of the
square base.
[Hint: use the volume formula to express the height of the box in terms of x.]
Simplify your formula as much as possible.
A(x) =
Next, find the derivative, A'(x).
= (x),V
Now, calculate when the derivative equals zero, that is, when A'(æ) = 0. [Hint: multiply both sides
by a².]
A' (x) = 0 when æ =
We next have to make sure that this value of x gives a minimum value for the surface area. Let's use
the second derivative test. Find A"(x).
A"(x) =
Evaluate A"(x) at the x-value you gave above.
NOTE: Since your last answer is positive, this means that the graph of A(x) is concave up around
that value, so the zero of A'(x) must indicate a local minimum for A(x). (Your boss is happy now.)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffe25867c-b393-4b1b-8545-ed6d3cde7ea9%2Fbdee4b08-1d5b-4522-b337-e13ef03754cf%2Fr7579ng_processed.png&w=3840&q=75)
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