1/2 points | Previous Answers SCalc8 3.2.504.XP. Does the function satisfy the hypotheses of the Mean Value Theorem on the given interval? f(x)=x+x-7, [0,2] There is not enough information to verify if this function satisfies the Mean Value Theorem. Yes, f is continuous on [0,2] and differentiable on (0, 2) since polynomials are continuous and differentiable on R. No, f is continuous on [0,2] but not differentiable on (0, 2). Yes, it does not matter if f is continuous or differentiable; every function satisfies the Mean Value Theorem. No, f is not continuous on [0,2]. My Notes Ask Your Teacher If it satisfies the hypotheses, find all numbers c that satisfy the conclusion of the Mean Value Theorem. (Enter your answers as a comma-separated list. If it does not satisfy the hypotheses, enter DNE). C = 0 Need Help? Read It Talk to a Tutor 10. 6/9 points | Previous Answers SCalc8 3.2.AE.003. 120 100 80 60 40 20 My Notes Ask Your Teacher EXAMPLE 3 To illustrate the Mean Value Theorem with a specific function, let's consider f(x)=x3x, a = 0, b = 4. Since f is a polynomial, it is continuous and differentiable for all x, so it is certainly continuous on [0, 4] and differentiable on (0, 4). Therefore, by the Mean Value Theorem, there is a number c in (0, 4) such that = f(4) f(0) f'(c)(4-0). Now (4) 60 becomes f(0) = 0 ' and f'(x) = 3x²-1 ' so this equation 60 Video Example = f'(c)(4) = 3c2-1 12c2-4 which gives 22 that is, c=±√4 But c must be in (0, 4), so c = 2 × . The figure illustrates this calculation: The tangent line at this value of c is parallel to the secant line. Need Help? Read It Talk to a Tutor -14 points SCalc8 3.7.011. My Notes Ask Your Teacher Consider the following problem: A farmer with 650 ft of fencing wants to enclose a rectangular area and then divide it into four pens with fencing parallel to one side of the rectangle. What is the largest possible total area of the four pens? (a) Draw several diagrams illustrating the situation, some with shallow, wide pens and some with deep, narrow pens. Find the total areas of these configurations. Does it appear that there is a maximum area? If so, estimate it. (b) Draw a diagram illustrating the general situation. Let x denote the length of each of two sides and three dividers. Let y denote the length of the other two sides. (c) Write an expression for the total area A in terms of both x and y. A = (d) Use the given information to write an equation that relates the variables. (e) Use part (d) to write the total area as a function of one variable. A(x) = (f) Finish solving the problem by finding the largest area. ft²