Calculus- Early Transcendentals, 2021a

5.3. The Mean Value Theorem 181
Exercises for Section 5.3
Exercise 5.3.1 Let f (x)=x 2 .Findavaluec∈ (−1,2) so that f ′ (c) equals the slope between the endpoints
of f (x) on [−1,2].
Exercise 5.3.2 Verify that f (x) =x/(x + 2) satisfies the hypotheses of the Mean Value Theorem on the
interval [1,4] and then find all of the values, c, that satisfy the conclusion of the theorem.
Exercise 5.3.3 Verify that f (x)=3x/(x + 7) satisfies the hypotheses of the Mean Value Theorem on the
interval [−2,6] and then find all of the values, c, that satisfy the conclusion of the theorem.
Exercise 5.3.4 Let f (x)=tanx. Show that f (π)= f (2π)=0 but there is no number c ∈ (π,2π) such that
f ′ (c)=0. Why does this not contradict Rolle’s theorem?
Exercise 5.3.5 Let f (x) =(x − 3) −2 . Show that there is no value c ∈ (1,4) such that f ′ (c) =(f (4) −
f (1))/(4 − 1). Why is this not a contradiction of the Mean Value Theorem?
Exercise 5.3.6 Describe all functions with derivative x 2 + 47x − 5.
Exercise 5.3.7 Describe all functions with derivative
1
1 + x 2 .
Exercise 5.3.8 Describe all functions with derivative x 3 − 1 x .
Exercise 5.3.9 Describe all functions with derivative sin(2x).
Exercise 5.3.10 Find f (x) if f ′ (x)=e −x and f (0)=2.
Exercise 5.3.11 Suppose that f is a differentiable function such that f ′ (x) ≥−3 for all x. What is the
smallest possible value of f (4) if f (−1)=2?
Exercise 5.3.12 Show that the equation 6x 4 −7x +1 = 0 does not have more than two distinct real roots.
Exercise 5.3.13 Let f be differentiable on R. Suppose that f ′ (x) ≠ 0 for every x. Prove that f has at most
one real root.
Exercise 5.3.14 Prove that for all real x and y |cosx − cosy| ≤|x − y|. State and prove an analogous
result involving sine.
Exercise 5.3.15 Show that √ 1 + x ≤ 1 +(x/2) if −1 < x < 1.
Exercise 5.3.16 Suppose that f (a)=g(a) and that f ′ (x) ≤ g ′ (x) for all x ≥ a.
(a) Prove that f (x) ≤ g(x) for all x ≥ a.