Using the Mean Value Theorem Quiz

The function f shown in the figure above is continuous on the closed interval [0,12] and differentiable on the open interval (0,12). Based on the graph, what are all values of x that satisfy the conclusion of the Mean Value Theorem applied to f on the closed interval [0,12]?

2 and 9 only because these are the values where the instantaneous rate of change of f at those values is equal to the average rate of change of f on [0,12].

Selected values of a differentiable function f are given in the table above. What is the fewest possible number of values of c in the interval [0,11] for which the Mean Value Theorem guarantees that f′(c)=4?

Two

Selected values of a differentiable function f are given in the table above. What is the fewest possible number of values of c in the interval [1,9] for which the Mean Value Theorem guarantees that f'(c) = 6?

Two

The Mean Value Theorem can be applied to which of the following functions on the closed interval [−3,3]?

f(x) = x-2/x-5

The Mean Value Theorem can be applied to which of the following functions on the closed interval [-5,5]?

f(x) = x^2/x^2-36

The function g shown in the figure above is continuous on the closed interval [0,x6] and differentiable on the open interval (0,x6), where x1, x2, x3, x4, x5, and x6 are points on the x-axis. Based on the graph, what are all values of x that satisfy the conclusion of the Mean Value Theorem applied to g on the closed interval [0,x6]?

x1 and x5 only, because these are the values where instantaneous rate of change of g at those values is equal to the average rate of change of g on [0,x6].