Your family drives 150 miles across Texas in exactly 2 hours, so your average speed is 75 mph. You never looked at the speedometer. Must it have read exactly 75 mph at some moment of the trip? And could a state trooper prove it without ever seeing your car at that moment?
In this section the derivative is used for something new. It does not calculate a number directly. It guarantees that a special point exists: a horizontal tangent, or an instant when the speed equals the average speed. We will come back to the road trip once we have the tool.
Rolle's Theorem
Start with the simplest picture. A function starts at some height and ends at the same height. In between, it must turn around, and at the turning point the tangent line is horizontal.
Why is it true? A function that is continuous on a closed interval reaches a maximum and a minimum there. If both occur at the endpoints, then $f$ is constant and $f' = 0$ everywhere. Otherwise one of them occurs inside the interval, and at an interior extreme of a differentiable function the tangent is horizontal.
Before you use the theorem, run this three-part check and write it down.
$f$ is continuous on the closed interval $[a, b]$.
$f$ is differentiable on the open interval $(a, b)$.
$f(a) = f(b)$. Compute both values yourself.
Sometimes the equal-endpoint condition is an equation you must solve for an unknown constant.
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Given the polynomial function $f(x) = x^2 - 6x + 8$, verify whether the hypotheses of Rolle's Theorem are satisfied on the interval $[2, 4]$. If so, find the value of…
Given the polynomial function $f(x) = x^2 - 6x + 8$, verify whether the hypotheses of Rolle's Theorem are satisfied on the interval $[2, 4]$. If so, find the value of $c \in (2, 4)$ whose existence is guaranteed by the theorem.
Consider the function $f(x) = x^2 - kx + 10$. Determine the value of the parameter $k \in \mathbb{R}$ so that the function satisfies the condition $f(1) = f(5)$ and…
Consider the function $f(x) = x^2 - kx + 10$. Determine the value of the parameter $k \in \mathbb{R}$ so that the function satisfies the condition $f(1) = f(5)$ and Rolle's Theorem can be applied on the interval $[1, 5]$. Once $k$ is found, find the value of $c$ where the derivative is zero.
Given the function $f(x) = \sin(x)$, verify that the hypotheses of Rolle's Theorem are satisfied on the interval $[0, \pi]$. Find the value of $c$ that satisfies the…
Given the function $f(x) = \sin(x)$, verify that the hypotheses of Rolle's Theorem are satisfied on the interval $[0, \pi]$. Find the value of $c$ that satisfies the conclusion of the theorem and represent it schematically on a unit circle or on the graph of the function.
All three theorems in this page start with the same two hypotheses, so checking them is the most important skill. Here is the quick guide.
Polynomials, $e^x$, $\sin x$ and $\cos x$ are continuous and differentiable everywhere.
Rational functions are continuous and differentiable wherever the denominator is not $0$.
$\ln x$ and $\sqrt{x}$ need $x > 0$. Always check that the whole interval lies inside the domain.
$|x|$ is continuous but has a corner at $0$, so it is not differentiable there.
Piecewise functions
At the point $x = k$ where the formula changes, you need two separate checks.
Continuity: the left piece and the right piece give the same value at $x = k$.
Differentiability: the two derivative formulas, evaluated at $x = k$, give the same number (equal one-sided derivatives). Equal values alone only give a continuous graph, and it could still have a corner.
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Explain with reasoning why Rolle's Theorem cannot be applied to the function $f(x) = 1 - |x-1|$ on the interval $[0, 2]$, even though $f(0) = f(2) = 0$. Which hypothesis…
Explain with reasoning why Rolle's Theorem cannot be applied to the function $f(x) = 1 - |x-1|$ on the interval $[0, 2]$, even though $f(0) = f(2) = 0$. Which hypothesis fails in this case?
Many problems leave a constant or an endpoint unknown and ask you to make Rolle's Theorem work. The method: turn each hypothesis into an equation.
Continuity at a junction gives one equation.
Equal one-sided derivatives at a junction give another.
$f(a) = f(b)$ gives a third.
Solve the system. (Here we call the unknowns $p, q, r$ so they are not confused with the point $c$.)
Choosing the interval
If the interval has an unknown endpoint, set $f(a) = f(b)$ and solve for it.
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Let the piecewise function be: $$f(x) = \begin{cases} x^2 + ax + b & \text{if } x \le 1 \\ \ln(x) + c & \text{if } x > 1 \end{cases}$$ Find the values of the parameters…
Let the piecewise function be:
$$f(x) = \begin{cases} x^2 + ax + b & \text{if } x \le 1 \\ \ln(x) + c & \text{if } x > 1 \end{cases}$$
Find the values of the parameters $a, b, c \in \mathbb{R}$ so that the function $f(x)$ satisfies the hypotheses of Rolle's Theorem on the interval $[0, e]$. Once the parameters are found, determine the value of $c \in (0, e)$ whose existence is guaranteed by the theorem.
Consider the function $f(x) = x^3 - 3x + m$. a) Determine for what values of the parameter $m$ the function satisfies the hypotheses of Rolle's Theorem on an interval of…
Consider the function $f(x) = x^3 - 3x + m$.
a) Determine for what values of the parameter $m$ the function satisfies the hypotheses of Rolle's Theorem on an interval of the form $[0, a]$ with $a > 0$.
b) For $m = 0$, find the largest possible interval $[p, q]$ where the theorem can be applied and find the point $c$ whose existence is guaranteed.
What if the endpoints are not at the same height? Then the graph can still be tilted so that a tangent line is parallel to the line joining the endpoints.
Rolle's Theorem is the special case where $f(a) = f(b)$: the secant is horizontal, so the slope is $0$.
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Apply the Mean Value Theorem to the function $f(x) = x^2 - x$ on the interval $[0, 2]$. Find the value of $c$ whose existence is guaranteed by the theorem and explain…
Apply the Mean Value Theorem to the function $f(x) = x^2 - x$ on the interval $[0, 2]$. Find the value of $c$ whose existence is guaranteed by the theorem and explain graphically the relationship between the tangent line at $c$ and the secant line passing through the endpoints of the interval.
Consider the function $f(x) = \sqrt{x^2 + 9}$. Find a point on the graph of the function where the tangent line is parallel to the secant line passing through the points…
Consider the function $f(x) = \sqrt{x^2 + 9}$.
Find a point on the graph of the function where the tangent line is parallel to the secant line passing through the points on the curve with x-coordinates $x=0$ and $x=4$. Which theorem guarantees that this point exists? State its hypotheses and verify that they are satisfied.
The MVT promises only one specific value: the slope $\dfrac{f(b) - f(a)}{b - a}$. It does not say a tangent line can have any slope you like.
To show that $f'$ takes some other value, use the Intermediate Value Theorem (IVT) on the function $f'$.
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It is known that a function $f(x)$ is continuous on the closed interval $[1, 5]$ and differentiable on the open interval $(1, 5)$. If the values of the function at the…
It is known that a function $f(x)$ is continuous on the closed interval $[1, 5]$ and differentiable on the open interval $(1, 5)$. If the values of the function at the endpoints are $f(1) = 10$ and $f(5) = 30$, can we ensure that there is a point $c$ in the interval where the slope of the tangent line is equal to $5$? Justify your answer using the Mean Value Theorem.
Let the function be $f(x) = x \cdot e^{x^2-1}$. a) Verify that $f(x)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0, 1]$. b) Is there any point…
Let the function be $f(x) = x \cdot e^{x^2-1}$.
a) Verify that $f(x)$ satisfies the hypotheses of the Mean Value Theorem on the interval $[0, 1]$.
b) Is there any point on this interval where the slope of the tangent line is equal to $e-1$? Justify your answer using the theorem.
If $s(t)$ is the position of a moving object, the average velocity over $[t_1, t_2]$ is the slope of a secant and the instantaneous velocity is $s'(t)$. The MVT turns into a statement about motion.
Back to the road trip. Over the 2-hour drive the average velocity was $\tfrac{150}{2} = 75$ mph. The position function is continuous and differentiable, so by the MVT there was an instant when the speedometer read exactly 75 mph. The same idea is why a toll road that records your entry and exit times can prove you were speeding even if no radar ever saw you.
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A marathon runner trains on a straight track. His or her position (in feet) relative to the starting point is given by the function $s(t) = t^2 + 2t$, where $t$ is the…
A marathon runner trains on a straight track. His or her position (in feet) relative to the starting point is given by the function $s(t) = t^2 + 2t$, where $t$ is the time in seconds. Use the Mean Value Theorem to prove that at some instant between $t=1$ and $t=4$ seconds, the runner's instantaneous velocity was exactly equal to the average velocity in that interval. Find this time.
An average speed camera on a highway detects that a vehicle enters the Lincoln Tunnel (point $A$) at 2:00:00 p.m. and exits it (point $B$) at 2:01:30 p.m. The distance…
An average speed camera on a highway detects that a vehicle enters the Lincoln Tunnel (point $A$) at 2:00:00 p.m. and exits it (point $B$) at 2:01:30 p.m. The distance between both control points is 3.5 miles.
If the speed limit in the tunnel is $120 \text{ mph}$, use the Mean Value Theorem to justify whether the driver should receive a speeding fine, regardless of whether their speed at the entrance and exit was legal.
A speed control system (average speed cameras) is located on a highway between mile markers 150 and 162. A vehicle enters the section at 10:00:00 and leaves at 10:06:40.…
A speed control system (average speed cameras) is located on a highway between mile markers 150 and 162. A vehicle enters the section at 10:00:00 and leaves at 10:06:40.
a) Using the Mean Value Theorem, justify whether the vehicle has exceeded the speed limit of $100 \text{ mph}$ at any time.
b) If the function describing the position of the vehicle in that section is $s(t) = 150 + 2t - 0.03t^2$ (where $s$ is in miles and $t$ in minutes since entering the section), find the exact time when the instantaneous velocity coincides with the average velocity of the journey.
Cauchy's theorem compares two functions at once. It is a generalization of the MVT.
Why $g'(x) \ne 0$? If $g(a) = g(b)$, Rolle's Theorem would give a point where $g' = 0$. So $g' \ne 0$ guarantees that the denominator $g(b) - g(a)$ is not $0$.
Geometric meaning. Think of $(g(t), f(t))$ as a point moving in the plane as $t$ goes from $a$ to $b$. The chord from the starting point to the ending point has slope $\dfrac{f(b) - f(a)}{g(b) - g(a)}$. The tangent line at parameter $t$ has slope $\dfrac{dy}{dx} = \dfrac{f'(t)}{g'(t)}$. Cauchy's theorem says that somewhere along the curve, a tangent is parallel to the chord.
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Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, verify whether the conditions of Cauchy's Theorem are satisfied on the interval $[1, 3]$. If so, find the value…
Given the functions $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, verify whether the conditions of Cauchy's Theorem are satisfied on the interval $[1, 3]$. If so, find the value of $c \in (1, 3)$ such that:
Apply Cauchy's Theorem to the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ on the interval $\left[ 0, \frac{\pi}{2} \right]$. Find the value of $c$ that satisfies the…
Apply Cauchy's Theorem to the functions $f(x) = \sin(x)$ and $g(x) = \cos(x)$ on the interval $\left[ 0, \frac{\pi}{2} \right]$. Find the value of $c$ that satisfies the equation:
Given the functions $f(x) = e^x$ and $g(x) = x^2 + 1$, apply Cauchy's Mean Value Theorem (Generalized Mean Value Theorem) on the interval $[0, 2]$. Find the value of…
Given the functions $f(x) = e^x$ and $g(x) = x^2 + 1$, apply Cauchy's Mean Value Theorem (Generalized Mean Value Theorem) on the interval $[0, 2]$.
Find the value of $c \in (0, 2)$, corresponding to point $P$ in the figure, that satisfies the conclusion of the theorem:
Rolle's Theorem can show that an equation $f(x) = 0$ has at most a certain number of solutions. The Intermediate Value Theorem shows it has at least one. Together they give "exactly one". (Some problems call the sign-change case of the IVT Bolzano's Theorem.)
Why? A smooth graph can only cross the same horizontal line twice by turning around in between, and every turn has a horizontal tangent.
To prove "exactly one root in $[a, b]$", show a sign change: $f(a)$ and $f(b)$ have opposite signs. Then show $f' \ne 0$ on $(a, b)$.
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Use Bolzano's Theorem and Rolle's Theorem to prove that the equation: $$3x^5 + 10x^3 + 15x - 4 = 0$$ has exactly one real solution on the interval $[0, 1]$.
Use Bolzano's Theorem and Rolle's Theorem to prove that the equation:
$$3x^5 + 10x^3 + 15x - 4 = 0$$
has exactly one real solution on the interval $[0, 1]$.
Prove, using both Bolzano's Theorem and Rolle's Theorem, that the equation: $$2x^5 + 5x^3 + 10x - k = 0$$ has a single real solution for any value of the constant…
Prove, using both Bolzano's Theorem and Rolle's Theorem, that the equation:
$$2x^5 + 5x^3 + 10x - k = 0$$
has a single real solution for any value of the constant $k \in \mathbb{R}$. Explain with reasoning why the existence of more than one root would contradict the statement of Rolle's Theorem.
Consider a function $f(x)$ differentiable on all of $\mathbb{R}$. It is known that the equation $f'(x) = 0$ has exactly two real solutions, $x_1 = -2$ and $x_2 = 3$.…
Consider a function $f(x)$ differentiable on all of $\mathbb{R}$. It is known that the equation $f'(x) = 0$ has exactly two real solutions, $x_1 = -2$ and $x_2 = 3$.
Prove that the function $f(x)$ cannot have more than three distinct real roots. Is it possible for the function to have only one real root? Justify your answer based on the geometric interpretations of the theorems of differentiability.
The MVT turns a statement about $f(b) - f(a)$ into a statement about $f'(c)$. If you can bound $f'$, you can bound the difference.
Comparing two functions
To compare two functions $f$ and $g$, apply the theorem to the difference $h = f - g$.
The same idea applies to motion: if two objects start at the same position and end at the same position at the same time, apply Rolle's Theorem to the difference of their positions. At some instant their velocities are equal.
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Use the Mean Value Theorem to prove the following inequality for any pair of real numbers $a$ and $b$ such that $0 < a < b$: $$\frac{b-a}{b} <…
Use the Mean Value Theorem to prove the following inequality for any pair of real numbers $a$ and $b$ such that $0 < a < b$:
Two runners, Ashley and Brandon, start a 100-meter hurdles race at the same time and reach the finish line in a dead heat (same time $T$). Their position functions are…
Two runners, Ashley and Brandon, start a 100-meter hurdles race at the same time and reach the finish line in a dead heat (same time $T$). Their position functions are $s_A(t)$ and $s_B(t)$, both continuous on $[0, T]$ and differentiable on $(0, T)$.
Prove, using Rolle's Theorem on the difference function $h(t) = s_A(t) - s_B(t)$, that there was at least one instant during the race when both runners had exactly the same instantaneous velocity.