A piece of cardboard measuring 12 inches by '10 inches is formed into an open-top box by cutting squares with side length 1' from each corner and folding up the sides. Find a formula for the volume of the box in terms of I Find the value for a: that will maximize the volume of the he): A box with a square base and open top must have a volume of 625W cma. We 1wish 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 I, the length of one side of the square base. [Hint: use the volume formula to express the height of the box in terms of I.] Simplify our formula as much as sible. I... =E Next, find the derivative, A (I). ,, .. =: Now, calculate when the derivative equals zero, that is, when A'EI) = U. [Hint: multiply both sides by :i:2 -] Allr} = Uwhen I =| We next have to make sure that this value of I gives a minimum value for the surface area. Let's use the second derivative test. Find .4' '(I. A m =:l Evaluate 1111') at the Ivalue you gave above. NOTE: Since your last answer is positive, this means that the graph of A(I) is concave up around that value, so the zero of A'EI} must indicate a local minimum for Alr}. {Your boss is happy now} Given the function g(x) = 6x - 54x- + 90x, find the first derivative, g'(x). g'(x) = 18x- - 108x + 90 Notice that g'(x) = 0 when x = 1, that is, g'(1) = 0. Now, we want to know whether there is a local minimum or local maximum at a = 1, so we will use the second derivative test. Find the second derivative, g"(x). g"(x) 36x - 108 Evaluate g"(1). g"(1) Based on the sign of this number, does this mean the graph of g( ) is concave up or concave down at * = 1? At x = 1 the graph of g(x) is Concave Down Based on the concavity of g(a) at a = 1, does this mean that there is a local minimum or local maximum at c = 1? At c = 1 there is a local MaximumAnswer the following questions for the function f(x) = IV I? + 25 defined on the interval 4 E I E 'II'. x) is concave down on the interval 1 I to x - (l V" a\" x) is concave up on the interval x - o" to x - |:] The inflection point for this function is at x - |:] The minimum for this function occurs at x - cl" w 0" The maximum for this function occurs at x . For -11