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## Homework Statement

a metal box with square base a no top holds 1000 cubic centimeters. it is formed by folding up the sides of the flattened pattern picture and seaming up the four sides. the material for the box costs $1.00 per square meter and the cost to seam the sides is 5 cents per meter. Find the dimensions of the box that costs the least to produce.

## The Attempt at a Solution

so for this problem ive managed to find the function the gives the cost to make the box

after converting everything to the same units my function is [tex]C(x)=x^2+\frac{.004}{x}+\frac{.0002}{x^2}[/tex]

so the derivative for the cost function is [tex]C'(x)=2x-\frac{.004}{x^2}-\frac{.0002}{x^3}[/tex]

setting this equal to zero and using maple, i get [itex]x=0.1494530180[/itex] plus 3 more zeros but one is negative and the other two are complex so in the context of this problem those 3 make no sense

after plotting my function i can see the critical point i have found is in fact the minimum, but im also asked to find the domain of my cost function and write up a proof that i have found a minimum and this is where im stuck.

## Homework Equations

if the cost function was given to me as just a regular function, one not in the context of this problem then i would say my domain would be all real except x=0, but I don't think it'll work for this

for my proof, i think ill have to use intermediate value theorem and extreme value theorem? but the extreme value theorem only works for closed intervals, and i still haven't found the domain