Suppose that f is continuous on [a, b] and differentiableon (a, b) and that m f(x) M on

In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. Let f(x)=[x], the greatest integer less than or equal to x. Then f (x)=0, so f(x) is constant by the Constant Function Theorem.

In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. If a<b and f (x) is positive on [a, b] then f(a) < f(b).

In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. If f(x) is increasing and differentiable on the interval [a, b], then f (x) > 0 on [a, b].

In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. The Racetrack Principle can be used to justify the statement that if two horses start a race at the same time, the horse that wins must have been moving faster than the other throughout the race.

In Exercises 15, decide if the statements are true or false. Give an explanation for your answer. Two horses start a race at the same time and one runs slower than the other throughout the race. The Racetrack Principle can be used to justify the fact that the slower horse loses the race.

Do the functions graphed in Exercises 69 appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? Do they satisfy the conclusion? 6

Do the functions graphed in Exercises 69 appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? Do they satisfy the conclusion? 7

Do the functions graphed in Exercises 69 appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? Do they satisfy the conclusion? 8

Do the functions graphed in Exercises 69 appear to satisfy the hypotheses of the Mean Value Theorem on the interval [a, b]? Do they satisfy the conclusion? 9

Applying the Mean Value Theorem with a = 2, b = 7 to the function in Figure 3.46 leads to c = 4. What is the equation of the tangent line at 4?

Applying the Mean Value Theorem with a = 3, b = 13 to the function in Figure 3.47 leads to the point c shown. What is the value of f (c)? What can you say about the values of f (x1) and f (x2)?

Let p(x) = x5+8x430x3+30x231x+22. What is the relationship between p(x) and f(x)=5x4 + 32x3 90x2 + 60x 31? What do the values of p(1) and p(2) tell you about the values of f(x)?

Let p(x) be a seventh-degree polynomial with 7 distinct zeros. How many zeros does p (x) have?

Use the Racetrack Principle and the fact that sin 0 = 0 to show that sin x x for all x 0.

Use the Racetrack Principle to show that ln x x 1.

Use the fact that ln x and ex are inverse functions to show that the inequalities ex 1+x and ln x x1 are equivalent for x > 0.

State a Decreasing Function Theorem, analogous to the Increasing Function Theorem. Deduce your theorem from the Increasing Function Theorem. [Hint: Apply the Increasing Function Theorem to f.]

Dominic drove from Phoenix to Tucson on Interstate 10, a distance of 116 miles. The speed limit on this highway varies between 55 and 75 miles per hour. He started his trip at 11:44 pm and arrived in Tucson at 1:12 am. Prove that Dominic was speeding at some point during his trip.

In Problems 1922, use one of the theorems in this section to prove the statements . If f (x) 1 for all x and f(0) = 0, then f(x) x for all x 0.

In Problems 1922, use one of the theorems in this section to prove the statements . If f(t) 3 for all t and f(0) = f (0) = 0, then f(t) 3 2 t 2 for all t 0.

In Problems 1922, use one of the theorems in this section to prove the statements . If f (x) = g (x) for all x and f(5) = g(5), then f(x) = g(x) for all x.

In Problems 1922, use one of the theorems in this section to prove the statements . If f is differentiable and f(0) < f(1), then there is a number c, with 0 <c< 1, such that f (c) > 0.

The position of a particle on the x-axis is given by s = f(t); its initial position and velocity are f(0) = 3 and f (0) = 4. The acceleration is bounded by 5 f(t) 7 for 0 t 2. What can we say about the position f(2) of the particle at t = 2?

Suppose that g and h are continuous on [a, b] and differentiable on (a, b). Prove that if g (x) h (x) for a<x<b and g(b) = h(b), then h(x) g(x) for a x b.

Deduce the Constant Function Theorem from the Increasing Function Theorem and the Decreasing Function Theorem. (See Problem 17.)

Prove that if f (x) = g (x) for all x in (a, b), then there is a constant C such that f(x) = g(x) + C on (a, b). [Hint: Apply the Constant Function Theorem to h(x) = f(x) g(x).]

Suppose that f (x) = f(x) for all x. Prove that f(x) = Cex for some constant C. [Hint: Consider f(x)/ex.]

Suppose that f is continuous on [a, b] and differentiable on (a, b) and that m f (x) M on (a, b). Use the Racetrack Principle to prove that f(x) f(a) M(x a) for all x in [a, b], and that m(x a) f(x) f(a) for all x in [a, b]. Conclude that m (f(b) f(a))/(b a) M. This is called the Mean Value Inequality. In words: If the instantaneous rate of change of f is between m and M on an interval, so is the average rate of change of f over the interval.

Suppose that f(x) 0 for all x in (a, b). We will show the graph of f lies above the tangent line at (c, f(c)) for any c with a<c<b. (a) Use the Increasing Function Theorem to prove that f (c) f (x) for c x<b and that f (x) f (c) for a<x c. (b) Use (a) and the Racetrack Principle to conclude that f(c) + f (c)(x c) f(x), for a<x<b.

In Problems 3032, explain what is wrong with the statement. The Mean Value Theorem applies to f(x) = |x|, for 2 <x< 2.

In Problems 3032, explain what is wrong with the statement. The following function satisfies the conditions of the Mean Value Theorem on the interval [0, 1]: f(x) =  x if 0 < x 1 1 if x = 0.

In Problems 3032, explain what is wrong with the statement. If f (x)=0 on a<x<b, then by the Constant Function Theorem f is constant on a x b.

In Problems 3337, give an example of: An interval where the Mean Value Theorem applies when f(x) = ln x.

In Problems 3337, give an example of: An interval where the Mean Value Theorem does not apply when f(x)=1/x.

In Problems 3337, give an example of: A continuous function f on the interval [1, 1] that does not satisfy the conclusion of the Mean Value Theorem.

In Problems 3337, give an example of: A function f that is differentiable on the interval (0, 2), but does not satisfy the conclusion of the Mean Value Theorem on the interval [0, 2].

In Problems 3337, give an example of: A function that is differentiable on (0, 1) and not continuous on [0, 1], but which satisfies the conclusion of the Mean Value Theorem.

Are the statements in Problems 3841 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (x) 0 for all x, then f(a) f(b) whenever a b.

Are the statements in Problems 3841 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (x) g (x) for all x, then f(x) g(x) for all x.

Are the statements in Problems 3841 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (x) = g (x) for all x, then f(x) = g(x) for all x.

Are the statements in Problems 3841 true or false for a function f whose domain is all real numbers? If a statement is true, explain how you know. If a statement is false, give a counterexample. If f (x) 1 for all x and f(0) = 0, then f(x) x for all x.