role of ministry of defence
Worksheet 21: The Mean Value Theorem
Russell Buehler
b.r@berkeley.edu
1. Verify that f (x) = x3 − x2 − 6x + 2 satisfies the hypotheses of Rolle’s theorem for the interval [0, 3], then find all c that satisfy the conclusion.
www.xkcd.com
2. Let f (x) = tan(x). Show that f (0) = f (π), but there is no number c in (0, π) such that f (c) = 0. Is this a counterexample to
Rolle’s theorem? Why or why not?
3. Verify that f (x) = x3 − 3x + 2 satisfies the hypotheses of the mean value theorem on [−2, 2], then find all c that satisfy the conclusion. 3
2
−x
4. Let f (x) = xx−1 on [0, 2]. Show that there is no value of c such that f (c) = value theorem? Why or why not?
f (2)−f (0)
.
2−0
Is this a counterexample to the mean
5. ( ) If for two functions f (x) and g(x), we know that f (x) = g (x) for every x in an interval (a, b), it must be the case that f − g is constant on (a, b). Why? What can we say about f (x) in terms of g(x)?
6. ( ) Using the mean value theorem and Rolle’s theorem, show that x3 + x − 1 = 0 has exactly one real root.
7. Show that the equation x4 + 4x + c = 0 has at most two real roots.
8. (a) Suppose that f is differentiable on R and has two roots. Show that f has at least one root.
(b) Suppose f is twice differentiable on R and has three roots. Show that f has at least one real root.
(c) Can you generalize parts (a) and (b)?
9. Label the local maxima/minima, absolute maximum/minimum, inflection points, and where the graph is concave up or concave down. 10. What is the first derivative test? What is the second derivative test?
11. For f (x) = 4x3 + 3x2 − 6x + 1, find the intervals on which f is increasing or decreasing, the local maxima and minima, and the inflection points.
Russell Buehler
b.r@berkeley.edu
1. Verify that f (x) = x3 − x2 − 6x + 2 satisfies the hypotheses of Rolle’s theorem for the interval [0, 3], then find all c that satisfy the conclusion.
www.xkcd.com
2. Let f (x) = tan(x). Show that f (0) = f (π), but there is no number c in (0, π) such that f (c) = 0. Is this a counterexample to
Rolle’s theorem? Why or why not?
3. Verify that f (x) = x3 − 3x + 2 satisfies the hypotheses of the mean value theorem on [−2, 2], then find all c that satisfy the conclusion. 3
2
−x
4. Let f (x) = xx−1 on [0, 2]. Show that there is no value of c such that f (c) = value theorem? Why or why not?
f (2)−f (0)
.
2−0
Is this a counterexample to the mean
5. ( ) If for two functions f (x) and g(x), we know that f (x) = g (x) for every x in an interval (a, b), it must be the case that f − g is constant on (a, b). Why? What can we say about f (x) in terms of g(x)?
6. ( ) Using the mean value theorem and Rolle’s theorem, show that x3 + x − 1 = 0 has exactly one real root.
7. Show that the equation x4 + 4x + c = 0 has at most two real roots.
8. (a) Suppose that f is differentiable on R and has two roots. Show that f has at least one root.
(b) Suppose f is twice differentiable on R and has three roots. Show that f has at least one real root.
(c) Can you generalize parts (a) and (b)?
9. Label the local maxima/minima, absolute maximum/minimum, inflection points, and where the graph is concave up or concave down. 10. What is the first derivative test? What is the second derivative test?
11. For f (x) = 4x3 + 3x2 − 6x + 1, find the intervals on which f is increasing or decreasing, the local maxima and minima, and the inflection points.