Problem 22 - Essential Calculus Early Transcendentals

Question

(a) Suppose that \(f\) is differentiable on \(\mathbb{R}\) and has two roots. Show that \(f^{\prime}\) has at least one root. (b) Suppose \(f\) is twice differentiable on \(\mathbb{R}\) and has three roots. Show that \(f^{\prime \prime}\) has at least one real root. (c) Can you generalize parts (a) and (b)?

Step-by-Step Solution

Answer

Use Rolle's Theorem to prove each part, and generalize by iteration.

Step 1: Understanding the Problem

Before starting, we need to understand the context of the problem. We are given a function, \(f\), with certain properties and asked to show that its derivatives must have roots under given conditions. Such results often use the Intermediate Value Theorem (IVT) and Rolle's Theorem.

Step 2: Part (a): Applying Rolle's Theorem

Since \(f\) is differentiable and has two roots, say at \(a\) and \(b\), \(f(a) = f(b) = 0\). Hence, \(f\) is continuous on \([a, b]\) and differentiable on \((a, b)\). By Rolle’s Theorem, there exists some \(c\) in \((a, b)\) such that \(f'(c) = 0\). This shows that \(f'\) has at least one root.

Step 3: Part (b): Understanding Again

In part (b), \(f\) is twice differentiable and has three roots. So, let's understand how more roots impact the second derivative. Again, Rolle's Theorem becomes crucial in this multi-step approach.

Step 4: Part (b): Applying Rolle's Theorem Twice

Suppose the three roots of \(f\) are \(a\), \(b\), and \(c\) with \(a < b < c\). By Rolle's Theorem applied on \([a, b]\) and \([b, c]\), \(f'\) has at least one root in each of \((a, b)\) and \((b, c)\). Let these roots be \(x_1\) and \(x_2\). Now, \(f'\) is differentiable on \([x_1, x_2]\), and by Rolle's Theorem again, there exists a point \(d\) in \((x_1, x_2)\) where \(f''(d) = 0\). This means \(f''\) has at least one root.

Step 5: Part (c): Generalizing the Results

The pattern observed can be generalized as follows: if \(f\) is \(n\)-times differentiable and has \((n+1)\) distinct roots, then the \(n\)-th derivative of \(f\) has at least one real root. This follows by applying Rolle's Theorem iteratively.

Key Concepts

Intermediate Value Theorem

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that ensures the existence of a value within a continuous interval based on boundary values. If you have a function, let's call it \( g(x) \), that is continuous on a closed interval \([a, b]\), then for any value \(N\) between \(g(a)\) and \(g(b)\), there exists at least one \(c\) in \([a, b]\) such that \(g(c) = N\).
This theorem is incredibly useful when verifying the existence of roots or specific values in a function's range.

• The function must be continuous on the interval you're considering.
• The desired value must be between the function's boundary values at the interval's endpoints.
• IVT does not tell us where the value occurs, just that it does.

It's important to understand IVT because it forms the basis of proving more advanced theorems.
This is especially crucial when you're dealing with derivatives, as continuity is key in many calculus applications.

Differentiable Functions

Differentiable functions are those for which the derivative exists at every point in their domain. If a function is differentiable on an interval, it is also continuous on that interval. This means that for a function to be differentiable, it smoothly changes without abrupt jumps or corners.
A differentiable function, let's say \( f(x) \), will have the property that you can find the derivative \( f'(x) \) everywhere in its domain.
Key aspects of differentiable functions include:

• There must not be any sharp corners or cusps where the slope would be undefined.
• It avoids vertical tangents where the function's slope becomes infinite.
• If \( f \) is differentiable on \([a, b]\), it's not only continuous but also follows the Mean Value Theorem conditions.

This concept is vital when dealing with Rolle's Theorem and for solving differential equations.
If you have a twice differentiable function, the second derivative \( f''(x) \) provides insight into the function's concavity and points of inflection.

Roots of Derivatives

A root of a derivative is a point where the derivative of a function equals zero, indicating a horizontal tangent line at that point. These roots signify that the original function could have a local maximum, local minimum, or a point of inflection. For instance, if \( f'(c) = 0 \), then \( c \) is a critical point of \( f(x) \).
To find these roots, you apply calculus theorems such as Rolle's Theorem. Rolle’s Theorem states that if a function is continuous on \([a, b]\), differentiable on \((a, b)\), and \( f(a) = f(b) \), then there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = 0 \).

• They provide critical points which help in understanding the function's behavior.
• These roots help in optimizing functions.
• Knowing the number of roots can give insight into the shape and extrema of the function.

Finding the roots of derivatives is a powerful tool in calculus for identifying where a function changes from increasing to decreasing or vice versa.

Calculus Theorems

Calculus theorems are crucial principles that guide the solutions and understanding of calculus problems. The notable theorems include the Intermediate Value Theorem, Mean Value Theorem, and Rolle's Theorem.
These theorems ensure the existence of certain points or values under specific conditions.

• Intermediate Value Theorem confirms that all values between the function's endpoints appear in the continuance of that function.
• Mean Value Theorem provides a point \( c \) in \([a, b]\) where the instantaneous rate of change (derivative) of a function matches the average rate of change over \([a, b]\).
• Rolle’s Theorem offers a certain location where the first derivative is zero, assuming the function reaches the same value at two different points.

These calculus theorems are indispensable in proving the existence of roots and understanding the behavior of differentiable functions.
Knowing and applying these theorems will be invaluable as you solve calculus problems related to the roots of functions and their derivatives.