Please help solve the following questions:

Assume that the function f is differentiable and increasing for all real numbers r, and the graph of f has exactly one inflection point. Of the following, which could be the graph of derivative f' of f? Justify your answers. (A) (B) fr . .X (C) (D) K. (E)The function f has a first derivative f'(x) = (x -1)3(x -3)?. (a) At what value(s) of r does f have a relative maximum? Justify your answers. (b) At what value(s) of r does f have a point of inflection? Justify your answers.For what value of k will f(x) = r + - have a relative maximum at 1 = -3? Justify your answers.Using the graphs provided, find the absolute minimum and maximum values of the given function on the given interval. If there is no maximum or minimum value, explain which part of the hypothesis of the Extreme Value Theorem is not satisfied. (a) [-1, 2] (b) [-1, 2] (c) (-1, 2) X 2 2 2Let f(x) = 315/3 - 15:2/3 Find the absolute maximum and minimum values of f(x) on [-1, 1].An open rectangular box with a square base is to be made from 48 square feet of material. What dimensions will result in a box with the largest possible volume?A particle is moving along the r-axis. Its position at time t 2 0 is given by I(t) = + + + 3t + 2 where c is measured in feet and t is measured in seconds. (a) Find all values of t > 0 for which the particle is moving to the left. (b) Find the total distance the particle travels during the first 6 seconds. (c) Is the particle speeding up or slowing time at t = 4? (d) Find the maximum speed of the particle for 0