Question

We are tasked with constructing a rectangular box with a volume of 1313 cubic feet. The material for the top costs 1010 dollars per square foot, the material for the 4 sides costs 22 dollars per square foot, and the material for the bottom costs 99 dollars per square foot. To the nearest cent, what is the minimum cost for such a box?

Answer 1

1st we have to find cost of box in terms of a single variable and then minimize it by using derivative

For relative maximum or minimum

To maximize or minimize a function f(x) 1st we need to find its critical points

For critical points we need to put differentiation of f(x) equal to zero i.e. f '(x) = 0

Then check second derivative of f(x) i.e. f ''(x) is positive or negative at critical numbers If f ''(x) >0 then that gives minimum value of f(x) and if f ''(x) <0 then that give maximum value of f(x)