Assignment:
Q1. An open-top box is to be made as follows: squares of a certain size will be cut away from each of the four corners of a 20" x 30" rectangle, and the ends will be folded upward to form the corner seams, as shown. How big should the square cutouts be in order to maximize the volume of the resulting box?
Q2. Let f(x) = ((x-2)2) / (1+x)
a. Use derivatives to identify all local maxima and/or minima of f, if any.
b. Use calculus to find the highest and lowest values attained by the function f on the interval 0 (greater than or equal to) x (less than or equal to) 5.
Q3. Let r(x) = (2/3)x3 - x - (1/3)x4 + (1/20)x5 + 1 :
Find the second derivative and set it equal to zero, in order to find inflection points.
Q4. Sketch continuous functions g and h satisfying the given conditions. Please sketch carefully enough to show the concavity clearly, and label axes and tickmarks where appropriate:
h ' (x) is negative for x < 1 and h' (x) is positive for x > 1; h" (x) is positive for
0 < x < 2, and h" (x) < 0 elsewhere
Provide complete and step by step solution for the question and show calculations and use formulas.