Answer all correctly please and show work. Preferably upload an image. Calculus.

2. The graph of the derivative function y = f'(x) is shown. a. On what intervals is the function f (x) increasing? 10 9 49 3 + A - - 1 0 b. At what x-values does the function f (x) have local maximum(s)? c. On what intervals is the function f (x) concave downward? d. State the x-coordinates of the points of inflection.3. Sketch the graph of a function f that is continuous on [1,5], has an absolute maximum at 5, absolute minimum at 2, local maximum at 3, local mimima at 2 and 4. 4. Explain the difference between an absolute minimum and a local minimum. 5. What theorem guarantees the existence of an absolute maximum value and an absolute minimum value for f, given f is a continuous function defined on a closed interval [a, b]? 6. Verify the function f (x) = 3x3 - 4x satisfies the hypotheses of the Mean Value Theorem on the interval [0, 3]. Then find all number(s) c that satisfy the conclusion of the Mean Value Theorem.7. The following function is continuous on the interval [0, 8] and f (0) = f(8), use this information and the graph below to estimate the value(s) of c that satisfy the conclusion of Rolle's Theorem on this interval. y = f( x ) 8. A box with a square base and open top must have a volume of 3200 cubic inches. We wish to find the dimensions of the box that minimize the amount of material used. a. Draw a picture and assign variables. b. What formula do you need to use to optimize? Are you looking for a maximum or a minimum value? c. What is your constraint equation? Use this equation to get your optimization equation in terms of one variable. d. Find any critical values that might lead to a max/min. e. What dimensions should the box be to achieve the desired results? Use the first derivative test to show you have found your max/min at this critical value. m