Problem 18 - University Calculus: Early Transcendentals

Question

Suppose that \(f^{\prime \prime}\) is continuous on \([a, b]\) and that \(f\) has three zeros in the interval. Show that \(f^{\prime \prime}\) has at least one zero in \((a, b)\) Generalize this result.

Step-by-Step Solution

Answer

If \( f \) has three zeros in \([a, b]\), then \( f'' \) has at least one zero in \((a, b)\). This result can be generalized: if \( f \) has \( n \) zeros, then \( f'' \) has at least \( n-2 \) zeros.

Step 1: Understanding the Problem

We need to show that if the second derivative of a function \( f \), which is continuous on \([a, b]\), has three zeros within the interval \((a, b)\), then there is at least one zero of the second derivative within \((a, b)\).

Step 2: Applying Rolle's Theorem

By Rolle's Theorem, if \( f(x) \) has three zeros, say \( x_1, x_2, \) and \( x_3 \) (where \( a < x_1 < x_2 < x_3 < b \)), there must be at least two distinct points \( c_1 \) and \( c_2 \) in \((x_1, x_3)\) where \( f'(c_1) = 0 \) and \( f'(c_2) = 0 \).

Step 3: Applying Rolle's Theorem Again

Since \( f'(c_1) = 0 \) and \( f'(c_2) = 0 \), and \( f' \) is continuous on \([c_1, c_2]\), by Rolle's Theorem again, there exists at least one point \( c_3 \) in \((c_1, c_2)\) such that \( f''(c_3) = 0 \). This shows that there is a zero of \( f'' \) in \((a, b)\).

Step 4: Generalization

If \( f \) has \( n \) zeros in \((a, b)\), \( f' \) must have at least \( n-1 \) zeros due to repeated application of Rolle's Theorem. Thus, \( f'' \) must have at least \( n-2 \) zeros. Each derivative reduces the number of zeros by one until \( f^{(n)} \) is considered, verifying the existence of zeros as required.

Key Concepts

Calculus

Calculus is a powerful branch of mathematics that focuses on change. It allows us to study the rates at which quantities change and to understand behaviors of functions through concepts like derivatives and integrals. Within calculus, derivatives help us grasp how a function changes at any given point. Whether it’s modeling the growth of populations or the speed of a moving object, calculus provides the tools.

When working with derivatives, it’s essential to grasp why these are calculated. They reveal important characteristics:

• First Derivative: Tells us the slope or rate of change of a function at a point.
• Second Derivative: Offers insight into the concavity of the function and helps us identify points of inflection or local maxima and minima.

These aspects are vital for understanding more complex problems that involve optimization and motion, establishing calculus as indispensable for fields like engineering and physics.

Second Derivative

The second derivative of a function, often denoted as \( f''(x) \), reveals much about the function's curvature and behavior. It helps determine the concave nature of the graph of a function. While the first derivative provides the slope or rate of change at a point, the second derivative gives us information on how that rate is changing.

Key points about the second derivative include:

• Concavity: If \( f''(x) > 0 \), the function is concave up (shaped like a cup), indicating a local minimum. Conversely, if \( f''(x) < 0 \), the function is concave down (shaped like a cap), indicating a local maximum.
• Points of Inflection: Where \( f''(x) = 0 \) or undefined, the function might switch concavity, which is known as an inflection point.

In the context of the problem, finding zeros of the second derivative impressed upon us that \( f''(c_3) = 0 \) implies such shifts. Noticing these changes is crucial in graph behavior, helping us predict trends or spot anomalies.

Continuity

Continuity in mathematics describes a function that is unbroken and without gaps. A continuous function means that small changes in the input (x-values) result in small changes in the output (y-values). This smoothness is crucial when applying concepts like Rolle's Theorem, which rely on an uninterrupted curve.

In our problem, the function \( f \) and its derivatives, particularly \( f'' \), are all continuous. That continuity is what ensures the applicability of Rolle's Theorem. The theorem states that if a continuous function achieves the same value at two distinct points, there must be at least one point within the interval where the first derivative is zero. Similarly, when applying the theorem repeatedly:

• Zero Derivative Bridge: Ensures the first and consequently the second derivative hits zero at some point between the function's zeros.

By understanding the seamless nature of continuity, students grasp the practical applications that enable them to find derivatives' zeros, thus exploring the deeper behavior of functions.