Problem 89 - Calculus Early Transcendentals

Question

Assume that \(f\) has at least two continuous derivatives on an interval containing \(a\) with \(f^{\prime}(a)=0 .\) Use Taylor's Theorem to prove the following version of the Second Derivative Test: a. If \(f^{\prime \prime}(x) > 0\) on some interval containing \(a,\) then \(f\) has a local minimum at \(a\) b. If \(f^{\prime \prime}(x) < 0\) on some interval containing \(a,\) then \(f\) has a local maximum at \(a\)

Step-by-Step Solution

Answer

Answer: The Second Derivative Test helps identify local minima and maxima of a function. It states that if the first derivative of a function is 0 and the second derivative is positive or negative on some interval, then the function has a local minimum or maximum at that point. Taylor's Theorem is used to prove this test by expressing the function as a Taylor series expansion and analyzing the function's behavior as based on the sign of the second derivative, ultimately demonstrating the existence of local minima or maxima.

Step 1: Write the Taylor Series Expansion around point a

Recall Taylor's Theorem, which states that: $$ f(x) = f(a) + f'(a)(x-a) + \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 + R_n(x) $$ where \(R_n(x)\) is the error term that goes to 0 as the higher-order derivatives approach 0. Since \(f^{\prime}(a)=0\), this simplifies to: $$ f(x) = f(a) + \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 + R_n(x) $$

Step 2: Prove local minimum case (a)

Assuming \(f^{\prime \prime}(x) > 0\) on some interval containing \(a\). Consider the Taylor expansion from Step 1 we have: $$ f(x) - f(a) = \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 + R_n(x) $$ As, \(f^{\prime \prime}(x) > 0\), we can safely say that \((x-a)^2\) is always positive, and therefore, the right side of the equation is positive. Adding \(f(a)\) to both sides, we have: $$ f(x) > f(a) $$ Thus, as \(f(x)\) is always greater than \(f(a)\) in the neighborhood of \(a\), this point has a local minimum at \(a\).

Step 3: Prove local maximum case (b)

Assuming \(f^{\prime \prime}(x) < 0\) on some interval containing \(a\). Similar to the minimum case, consider the Taylor expansion from Step 1: $$ f(x) - f(a) = \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 + R_n(x) $$ As, \(f^{\prime \prime}(x) < 0\), we can safely say that \((x-a)^2\) is always positive, and therefore, the right side of the equation is negative. Adding \(f(a)\) to both sides, we have: $$ f(x) < f(a) $$ Thus, as \(f(x)\) is always less than \(f(a)\) in the neighborhood of \(a\), this point has a local maximum at \(a\). We have now proven both the cases using Taylor's Theorem, and thus the Second Derivative Test has been demonstrated.

Key Concepts

Taylor's Theorem

Taylor's Theorem is a foundational concept in calculus used to approximate the value of a function near a point using the function's derivatives at that point. It essentially creates a polynomial that mirrors the original function's shape locally, which allows for analysis of the function's behavior without extensively solving complex equations.

The theorem provides us with the following formula:

• \( f(x) = f(a) + f'(a)(x-a) + \frac{1}{2} f^{\prime \prime}(a)(x-a)^2 + R_n(x) \)

Here, \( R_n(x) \) is the remainder term, which diminishes as higher-order derivatives get small, enabling more accurate approximations.

In the specific example of identifying local minimums and maximums, we are particularly interested in the terms up to \( f^{\prime \prime}(a) \), because they can dictate the concavity and thus the nature of the points. Thus, Taylor's Theorem lays the groundwork for the Second Derivative Test.

Local Minimum

A local minimum of a function is a point where the function value is smaller than at any other point in its immediate vicinity. Understanding local minimums is crucial for optimizing processes and finding 'low points' of functions.

To determine if a point \(a\) is a local minimum using the Second Derivative Test, we first ensure \(f^{\prime}(a) = 0\). After this, if the second derivative \(f^{\prime \prime}(x) > 0\) in a neighborhood around \(a\), then the function \(f\) is upwards concave at point \(a\), implying \(f(a)\) is indeed a local minimum.

Mathematically, this translates to losing track of decreasing values as we move away from \(a\), thereby establishing \(a\) as a low point locally. This is crucial for analyses requiring minimization.

Local Maximum

A local maximum refers to a point on a function where the value is greater than at any other nearby point. Discovering these peaks is important in fields like economics and physics, where we want to identify potential high points efficiently.

When applying the Second Derivative Test, once a point \(a\) satisfies \(f^{\prime}(a) = 0\), further examination of \(f^{\prime \prime}(x) < 0\) in a surrounding interval confirms a local maximum. The function demonstrates this by showing a downward concave shape at \(a\), making \(f(a)\) the highest value in its vicinity.

Thus, the study of concavity through the second derivative helps reveal points of peak function values without exhaustive searches across all possible values.

Continuous Derivatives

Continuous derivatives are the derivatives that do not have abrupt changes or interruptions in their values. Having two continuous derivatives is a fundamental requirement for applying the Second Derivative Test.

Why is this important? Because continuity ensures smooth transitions in the behavior of the function, allowing us to accurately predict local behavior around point \(a\).

For Taylor's Theorem to be applied effectively, the smoothness granted by continuous derivatives means the approximations generated will be reliable and insightful.

Simply put, with continuous derivatives, predictions regarding minima, maxima, and points of inflection can confidently be based on the function’s immediate derivative behavior without unexpected shifts.