Problem 17 - Calculus - AP Edition

Question

(a) Show that if \(f\) is differentiable on \((-\infty,+\infty),\) and if \(y=f(x)\) and \(y=f^{\prime}(x)\) are graphed in the same coordinate system, then between any two \(x\) -intercepts of \(f\) there is at least one \(x\) -intercept of \(f^{\prime}\) (b) Give some examples that illustrate this.

Step-by-Step Solution

Answer

Rolle's Theorem ensures at least one intercept of \( f'(x) \) between any two intercepts of \( f(x) \). Examples like \( f(x)=x^3-3x \) confirm this.

Step 1: Define the Problem

We are given that the function \( f \) is differentiable over the interval \((-$\infty,+\infty)\). We need to prove that if you graph \( f(x) \) and its derivative \( f'(x) \) on the same axes, between any two \( x \)-intercepts of \( f(x) \), there must be an \( x \)-intercept of \( f'(x) \).

Step 2: Use Rolle's Theorem

Rolle's Theorem states that if a function \( g(x) \) is continuous on the closed interval \([a, b]\), differentiable on the open interval \((a, b)\), and \( g(a) = g(b) = 0 \), then there exists at least one \( c \in (a, b) \) such that \( g'(c) = 0 \).

Step 3: Apply Rolle's Theorem to \( f(x) \)

Consider two \( x \)-intercepts of \( f(x) \), say \( x_1 \) and \( x_2 \) where \( f(x_1) = 0 \) and \( f(x_2) = 0 \). As \( f(x) \) is continuous and differentiable, by Rolle's Theorem, there exists a point \( c \in (x_1, x_2) \) where \( f'(c) = 0 \). Thus, between any two \( x \)-intercepts of \( f(x) \), there is at least one \( x \)-intercept of \( f'(x) \).

Step 4: Example 1

Consider the function \( f(x) = x^3 - 3x \). This function has \( x \)-intercepts at \( x = -\sqrt{3}, 0, \sqrt{3} \). Its derivative is \( f'(x) = 3x^2 - 3 \), which has an \( x \)-intercept at \( x = \pm 1 \). Between \(-\sqrt{3}\) and \(0\), \( x = -1 \) is an intercept of \( f'(x) \). Similarly, between \(0\) and \(\sqrt{3}\), \( x = 1 \) is an intercept of \( f'(x) \).

Step 5: Example 2

Consider the function \( f(x) = x^3 - x \). This function has \( x \)-intercepts at \( x = -1, 0, 1 \). Its derivative is \( f'(x) = 3x^2 - 1 \), which has an \( x \)-intercept at \( x = \pm \sqrt{\frac{1}{3}} \). Between \(-1\) and \(0\), \( x = -\sqrt{\frac{1}{3}} \) is an intercept of \( f'(x) \). Similarly, between \(0\) and \(1\), \( x = \sqrt{\frac{1}{3}} \) is an intercept of \( f'(x) \).

Key Concepts

Differentiability

Differentiability refers to a property of a function that allows it to have a well-defined tangent at every point in its domain. For a function to be differentiable, it must be smooth and continuous.

• If a function is differentiable at a point, it means you can calculate the derivative at that point.
• Differentiability implies that there are no sharp edges or breaks in the graph of the function.
• A differentiable function is continuous, but not all continuous functions are differentiable. A classic example is the absolute value function, which is continuous everywhere but not differentiable at the point where it changes direction.

Rolle's Theorem, a key player in the original exercise, relies on the concept of differentiability. If a function is differentiable across an interval, the theorem assures that under certain conditions, there's at least one point within that interval where the derivative equals zero, signifying a horizontal tangent.

Differentiability is crucial in calculus as it lays the foundation for many theorems and methods, including those involving the calculation and interpretation of slopes and changes in graphs.

X-intercepts

In graphing, an x-intercept is the point where a graph crosses the x-axis. This means the value of the function at these intercepts equals zero.

• Finding x-intercepts aligns with determining the roots or solutions to the equation given by a function set to zero.
• A graph can have multiple x-intercepts or sometimes none, depending on the nature of the function.
• In the context of polynomials, x-intercepts are often referred to as roots or zeros.

In the exercise, we were interested in the x-intercepts of two related functions, a function and its derivative. When applying Rolle's Theorem, identifying x-intercepts helps pinpoint intervals where we expect the derivative to also have x-intercepts.

Understanding where a function crosses the x-axis helps determine intervals of increase or decrease and is crucial when analyzing the behavior of functions.

Derivative Function

A derivative is a powerful tool in calculus that provides the rate at which a function changes at any given point. It forms the basis of various applications, including optimization and curve sketching.

• The derivative function, often denoted as \( f'(x) \), represents a new function derived from the original function \( f(x) \).
• This derivative function assesses how the parent function increases or decreases, showing slopes of tangents at various points.
• Zero values of this derivative function represent stationary points, which can be potential maxima, minima, or inflection points.

In the context of the exercise, the derivative function is crucial. By noticing where \( f'(x) = 0 \), we could show that between any two x-intercepts of \( f(x) \), there must be an x-intercept of \( f'(x) \). This illustrates how derivative functions reveal turning points and intervals where functions switch direction.