Problem 13 - Thomas Calculus

Question

Show that if \(f^{\prime \prime}>0\) throughout an interval \([a, b],\) then \(f^{\prime}\) has at most one zero in \([a, b] .\) What if \(f^{\prime \prime}<0\) throughout \([a, b]\) instead?

Step-by-Step Solution

Answer

If \(f'' > 0\), \(f'\) has at most one zero; similarly, if \(f'' < 0\), \(f'\) has at most one zero.

Step 1: Understanding the Concept of Concavity

When the second derivative \(f''(x) > 0\), the function \(f(x)\) is concave upwards on the interval \([a, b]\). This implies that the slope \(f'(x)\) is increasing throughout the interval. Conversely, if \(f''(x) < 0\), the function is concave downwards, indicating that \(f'(x)\) is decreasing.

Step 2: Analyzing the Zero of the First Derivative

Since \(f'(x)\) is strictly increasing when \(f''(x) > 0\), \(f'(x)\) can cross the x-axis at most once. Thus, \(f'(x)\) has at most one zero on \([a, b]\). On the other hand, if \(f''(x) < 0\), \(f'(x)\) is strictly decreasing, so it can also have at most one zero in \([a, b]\).

Step 3: Graphical Representation to Justify

By visualizing the graph: for \(f''(x) > 0\), \(f'(x)\) starts at a lower value and increases, potentially crossing zero once. If it were to cross zero more than once, it would contradict the increasing nature. Similarly, for \(f''(x) < 0\), \(f'(x)\) might start at a higher value and decrease, potentially crossing zero once.

Step 4: Conclusion Based on Analysis

The behavior of the derivative \(f'(x)\) is determined by \(f''(x)\). If \(f''(x) > 0\), \(f'(x)\) is increasing and crosses zero at most once. If \(f''(x) < 0\), \(f'(x)\) is decreasing and also crosses zero at most once.

Key Concepts

Concavity

In calculus, concavity refers to the way a curve bends. The second derivative of a function tells us about this bending, or the curve's concavity. If the second derivative, denoted as \( f''(x) \), is greater than zero, the function is said to be concave upwards on a certain interval. This means that the curve is shaped like a "U".
On the other hand, if \( f''(x) \) is less than zero, the function is concave downwards and looks like an upside-down "U".

• Concave Upwards: \( f''(x) > 0 \) - the function curves upwards like a smile.
• Concave Downwards: \( f''(x) < 0 \) - the function curves downwards like a frown.

Understanding concavity helps visualize how a function behaves over a particular interval. It also allows us to infer the behavior of its first derivative, \( f'(x) \), which is the slope or gradient of the function.

Second Derivative

The second derivative, \( f''(x) \), plays a crucial role in understanding a function's shape and behavior. It is the derivative of the first derivative \( f'(x) \), and it provides information about the rate of change of \( f'(x) \).
When we know that \( f''(x) > 0 \), \( f'(x) \) is an increasing function. Essentially, this means that the slope itself is becoming steeper as you move along the curve.

• If \( f''(x) > 0 \), \( f'(x) \) increases, showing an upward curvature.
• If \( f''(x) < 0 \), \( f'(x) \) decreases, indicating a downward curvature.

This concept is especially useful for predicting where the function might have local minima or maxima, and it assists in determining the nature of turning points.

Zero of Derivative

The zero of a derivative, \( f'(x) = 0 \), represents points where the function's slope is horizontal. These are critical points where the function might achieve a local maximum or minimum.
By analyzing \( f''(x) \), we determine the nature or occurrence of these zeros.

• With \( f''(x) > 0 \), \( f'(x) \) is increasing, crossing zero at most once. Thus, the function changes direction at most one time in this interval.
• Similarly, with \( f''(x) < 0 \), \( f'(x) \) is decreasing and can also cross zero only once.

The zeros of the derivative mark potential transition points in the behavior of the original function. Knowing the concavity assists in projecting the graph's movements at these points, solidifying our understanding of the function's behavior over an interval.