Question

suppose you want to make an open topped box out of 4x6 index card by cutting a square out of each corner and then folding the edges. how large a square should you cut out of each corner in order to maximize the volume of the resulting box

Answer 1

so check the picture below

notice, since you're cutting out a square, the sides must all be equal, thus the largest "x" can't be 2, half of 4, it has to be just less than 2 or it has no volume, so x<2, and can't be 0, because, you'd have no volume either, so x>0, so 0<x<2

$\bf V=(4-2x)(x)(6-2x)\implies V=(4x-2x^2)(6-2x) \\\\\\ V=4x^3-20x^2+24x \\\\\\ \cfrac{dv}{dx}=12x^2-40x+24\implies 0=12x^2-40x+24 \\\\\\ 0=3x^2-10x+6$

anyway... so that'd be dv/dx... you can just run it through the quadratic formula to get the critical points, and run a first-derivative test on them, bearing in mind the range for "x",  (0, 2)