Problem 47 - Thomas Calculus

Question

The (second) second derivative test Use the equation $$f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime \prime}\left(c_{2}\right)}{2}(x-a)^{2}$$ to establish the following test. \begin{equation} \begin{array}{l}{\text { Let } f \text { have continuous first and second derivatives and }} \\ {\text { suppose that } f^{\prime}(a)=0 . \text { Then }} \\ {\text { a. } f \text { has a local maximum at } a \text { if } f^{\prime \prime} \leq 0 \text { throughout an interval }} \\ {\text { whose interior contains } a ;} \\ {\text { b. } f \text { has a local minimum at } a \text { if } f^{\prime \prime} \geq 0 \text { throughout an interval }} \\\ {\text { whose interior contains a. }}\end{array} \end{equation}

Step-by-Step Solution

Answer

The second derivative test shows: if \( f^{\prime\prime} \leq 0 \), \( f \) has a local max; if \( f^{\prime\prime} \geq 0 \), a local min.

Step 1: Understanding the Expansion

The given formula \( f(x)=f(a)+f^{\prime}(a)(x-a)+\frac{f^{\prime\prime}\left(c_{2}\right)}{2}(x-a)^{2} \) is a Taylor expansion centered at \( a \). It approximates the function \( f(x) \) near the point \( x = a \) using the values of the function and its derivatives at \( a \).

Step 2: Applying the Condition \( f^{\prime}(a) = 0 \)

We are given that \( f^{\prime}(a) = 0 \). Substituting this into the Taylor expansion equation simplifies it to: \[ f(x) = f(a) + \frac{f^{\prime\prime}\left(c_{2}\right)}{2}(x-a)^{2} \]. Here, \( f^{\prime}(a)(x-a) \) vanishes, as \( f^{\prime}(a) = 0 \).

Step 3: Suppose \( f^{\prime\prime} \leq 0 \)

Consider the case where \( f^{\prime\prime} \leq 0 \) throughout an interval containing \( a \). In the simplified expansion \( f(x) = f(a) + \frac{f^{\prime\prime}(c_{2})}{2}(x-a)^{2} \), if \( f^{\prime\prime}(c_{2}) \leq 0 \), then the term \( \frac{f^{\prime\prime}(c_{2})}{2}(x-a)^{2} \) is zero or negative for all \( x eq a \). This implies \( f(x) \leq f(a) \), suggesting \( f \) has a local maximum at \( a \).

Step 4: Suppose \( f^{\prime\prime} \geq 0 \)

Now consider \( f^{\prime\prime} \geq 0 \) throughout an interval containing \( a \). The term \( \frac{f^{\prime\prime}(c_{2})}{2}(x-a)^{2} \), in this case, is zero or positive, meaning \( f(x) \geq f(a) \) for all \( x eq a \). This suggests \( f \) has a local minimum at \( a \).

Step 5: Conclusion

Thus, when \( f^{\prime}(a) = 0 \): if \( f^{\prime\prime} \leq 0 \), \( f \) has a local maximum; if \( f^{\prime\prime} \geq 0 \), \( f \) has a local minimum.

Key Concepts

Taylor Expansion

Taylor expansion is a powerful tool used to approximate the value of a function near a specific point. Imagine you have a bumpy road, and you're trying to describe a small section of it. Taylor expansion helps by taking a function and expanding it into a polynomial, which is usually easier to work with.
This expansion uses derivatives of the function at the point where the expansion is centered, often denoted as point 'a'.

• The first derivative, \( f'(a) \), describes the slope or the tilt of the function at 'a'.
• The second derivative, \( f''(a) \), describes how the slope is changing at that point, giving insight into the curvature.

By retaining more terms in the expansion, higher accuracy near the chosen point can be achieved. The simplified Taylor expansion, given a specific condition like \( f'(a) = 0 \), focuses on the quadratic component to analyze the nature of the function close to 'a'.

Local Maximum

A local maximum refers to a point where a function reaches a peak in its immediate area, meaning the function's value at that point is larger than or equal to any nearby points.
Suppose we explore whether point 'a' is a local maximum using the second derivative test. If \( f'(a) = 0 \) and the second derivative \( f'' \leq 0 \) for the whole vicinity around 'a', then this means the function curves downwards at this point.

• The zero slope condition \( f'(a) = 0 \) means the function has a flat tangent line at 'a'.
• If \( f''(c_2) \leq 0 \), the function is either shaping downward or flat, confirming a peak.
• This aligns with our intuition that a downward opening parabola indicates a summit or peak at its vertex.

Thus, the test concludes there is a local maximum at 'a'.

Local Minimum

Conversely, a local minimum is the concept's yang to the local maximum's yin. It represents a point where the function's value is the smallest within its neighborhood. If you're using the second derivative test, this involves checking the curvature of the function, much like the verification of a local maximum.

• Start by confirming that \( f'(a) = 0 \), ensuring a flat tangent line at 'a'.
• If then \( f''(c_2) \geq 0 \), this reveals that the function curves upwards at 'a'.
• The upward curvature, like a happy smile, suggests a valley or trough at that section.

The test therefore deduces that 'a' is a local minimum, edging out all other nearby points in its decreased value.

Function Derivatives

Function derivatives are fundamental tools in calculus that help us understand how a function changes. They're like a magnifying glass, peering closely at functions to reveal their secrets.

• The first derivative, \( f'(x) \), tells us about the slope or rate of change of the function at any point. It's crucial for identifying stationary points where the slope becomes zero.
• The second derivative, \( f''(x) \), gives insight into the function's concavity, indicating whether it is curving upwards or downwards.

When you pair these together, they provide a powerful technique for analyzing the behavior of functions.
Knowing these can help, for example, in graphing, optimization problems, and determining crucial traits like local maxima and minima, making derivatives indispensable in mathematics.