Problem 11 - Basic Analysis: Introduction to Real Analysis

Question

Prove the more general version of the second derivative test. Suppose \(f:(a, b) \rightarrow \mathbb{R}\) is differentiable and \(x_{0} \in(a, b)\) is such that, \(f^{\prime}\left(x_{0}\right)=0, f^{\prime \prime}\left(x_{0}\right)\) exists, and \(f^{\prime \prime}\left(x_{0}\right)>0 .\) Prove that \(f\) has a strict relative minimum at \(x_{0} .\) Hint: Consider the limit definition of \(f^{\prime \prime}\left(x_{0}\right) .\)

Step-by-Step Solution

Answer

Given conditions ensure a strict relative minimum at \(x_0\).

Step 1: Understanding the Problem Statement

We need to prove that if a function \(f\) satisfies certain conditions at a point \(x_0\), it has a strict relative minimum there. Specifically, \(f^{\prime}(x_0) = 0\) and \(f^{\prime\prime}(x_0) > 0\). Use the second derivative \(f^{\prime\prime}(x_0)\) and its positivity to deduce the behavior of \(f\) near \(x_0\).

Step 2: Consideration of the Second Derivative Concept

The second derivative, \(f^{\prime\prime}(x_0)>0\), indicates that the graph of \(f\) is convex near \(x_0\). This implies that near \(x_0\), \(f(x)\) behaves like a convex parabola opening upward.

Step 3: Applying the Limit Definition

The limit definition of the second derivative, \(f^{\prime \prime}(x_0) = \lim_{h \to 0}\frac{f^{\prime}(x_0 + h) - f^{\prime}(x_0)}{h}\), and knowing \(f^{\prime}(x_0) = 0\), simplifies to \(f^{\prime}(x_0 + h) / h > 0\) for small \(h\).

Step 4: Implication of Positive Second Derivative

Because \(f^{\prime \prime}(x_0) > 0\), for small enough \(h > 0\), \(f^{\prime}(x_0 + h) > 0\) and for \(h < 0\), \(f^{\prime}(x_0 + h) < 0\). This suggests that \(f\) decreases as \(x\) approaches \(x_0\) from the left and increases as \(x\) departs \(x_0\) to the right, forming a minimum.

Step 5: Concluding the Minimum Existence

Since \(f\) is decreasing towards \(x_0\) and increasing past \(x_0\), \(x_0\) must be a point where \(f\) attains a strict relative minimum, confirming the condition for a minimum based on the behavior of the first and second derivatives.

Key Concepts

Differentiability

A function's differentiability is crucial for analyzing its behavior around specific points. A function is differentiable at a point if it has a derivative at that point. In simpler terms, it means that the function has a well-defined tangent line at the location in question, allowing us to understand how the function changes nearby. Differentiable functions are smooth, without any breaks, sharp turns, or cusps at the points of differentiability. This smoothness is essential for using calculus tools like derivatives to predict and analyze function behavior.

• The function must be continuous at the point.
• The derivative must exist at that point.

If these conditions are met, we can utilize derivatives effectively, which is key when applying tests like the second derivative test.

Strict Relative Minimum

A strict relative minimum is an important concept in calculus that refers to a point where the function's value is lower than at any nearby points. This concept connects closely with the behavior and geometrical properties of functions. Critical conditions for a strict relative minimum involve the first and second derivatives:

• The first derivative at the point equals zero, indicating that the slope of the tangent line is flat.
• The second derivative is positive, suggesting a convex shape or a curve opening upwards.

These conditions suggest that the function decreases toward the point and increases away, making the point a minimum within its vicinity. Understanding these conditions helps in identifying and confirming strict relative minimum points through analytical methods.

Convexity

Convexity is a fundamental property in the study of functions, especially in optimization and calculus. A function is convex if, for any two points on its graph, the line segment connecting them lies above or on the graph. In mathematical terms, if the second derivative of a function at a point is positive, the function is convex at that region. This implies several important behaviors:

• The function has a bowl shape at that point, meaning no peaks or valleys within that interval.
• Convex functions have only one local minimum which is also a global minimum within the neighborhood.

Convexity helps determine the nature of stationary points, evidenced by the second derivative test, indicating the potential for a minimum when the second derivative is positive.

Limit Definition

The limit definition of derivative is a cornerstone in calculus, providing a precise way of finding the rate of change of functions. It formally defines how derivatives are calculated using a limit process.For a function to have a second derivative, the limit definition is extended to the second order of differentiation:\[ f^{\prime \prime}(x_0) = \lim_{h \to 0} \frac{f^{\prime}(x_0 + h) - f^{\prime}(x_0)}{h} \]Where:

• \( h \) is a small increment to \( x_0 \)
• The numerator \( f^{\prime}(x_0 + h) - f^{\prime}(x_0) \) represents the change in the first derivative over \( h \)

This definition is key for understanding the concavity or convexity at a point by telling us whether the rate of the rate of change is increasing or decreasing. A positive second derivative suggests that the rate of change itself is increasing, confirming a convex shape.