Chapter 3.1, Problem 2E

The function f(x) is everywhere continuous and differentiable. Prove that if the function f(x) has two real roots, then its derivative f′(x) has at least one root.

The pounds of bananas sold each week at all Metro Seattle Albertsons stores as a function of price, p , in dollars/pound(lb.) is given by q(p) = 100e7.5-1.5p1. Express the Revenue function in terms of p and then find both the first and second derivatives of the revenue function. Type each of these in your text box, using appropriate standard mathematical formatting notation (like that shown above) in Excel. 2. Use Excel over an interval of [0, 6] in increments of .25 to create values for all 3 of your functions from part 1. This is hard! You should be getting the beginning values shown on the next page. Keep at it until you do!! You must create these values by typing in and using the correct formulas. 3. Use Excel to determine exactly where there are any Maximum and/or Minimum values for Revenue. In your text box, explain fully and completely how you determine where to look, and how you know from the First Derivative Test that you have a maximum or a minimum value. 4. Write a…

use definition to show that the function f(x)= |x|^3 is differentiable at 0, with derivative equal to 0.