Problem 64 - Calculus Early Transcendentals: Pearson New International Edition

Question

Recall that the Second Derivative Test for Local Extrema (Section 4.3) does not apply when \(f^{\prime \prime}(c)=0\). Prove the following generalization, which may help determine a maximum or a minimum when \(f^{\prime \prime}(c)=0 .\) Suppose that $$ f^{\prime}(c)=f^{\prime \prime}(c)=f^{\prime \prime \prime}(c)=\cdots=f^{(n)}(c)=0 $$ where \(n\) is odd and \(f^{(n+1)}(x)\) is continuous near \(c\). 1\. If \(f^{(n+1)}(c)<0\), then \(f(c)\) is a local maximum value. 2\. If \(f^{(n+1)}(c)>0\), then \(f(c)\) is a local minimum value. Test this result on \(f(x)=x^{4}\).

Step-by-Step Solution

Answer

For \( f(x) = x^4 \), \( x = 0 \) is a local minimum.

Step 1: Define the Function and Calculate Its Derivatives

We are tasked to test the given result on the function \( f(x) = x^4 \). Begin by defining the function and calculating its derivatives. - First derivative: \( f'(x) = 4x^3 \) - Second derivative: \( f''(x) = 12x^2 \) - Third derivative: \( f'''(x) = 24x \) - Fourth derivative: \( f^{(4)}(x) = 24 \) - Fifth derivative: \( f^{(5)}(x) = 0 \)Here, the derivatives must be calculated up to the point where they are non-zero at the point of interest which is \( x=0 \).

Step 2: Evaluating Derivatives at x = 0

Now, evaluate these derivatives at \( x = 0 \) to determine the behavior of \( f(x) \) at this point.- \( f'(0) = 4 \cdot 0^3 = 0 \)- \( f''(0) = 12 \cdot 0^2 = 0 \)- \( f'''(0) = 24 \cdot 0 = 0 \)- \( f^{(4)}(0) = 24 \)- \( f^{(5)}(0) = 0 \)All derivatives up to \( f^{(4)}(x) \) are zero at \( x = 0 \), and the fourth derivative is positive.

Step 3: Applying the Generalization for Local Extrema

Given that all derivatives up to the nth order are zero, and \( f^{(n+1)}(x) \) is non-zero, we can apply the generalization.Here, \( n = 3 \) because the first non-zero derivative after the zeros is the 4th derivative which is 24 (\( f^{(4)}(0) = 24 \)), and it is positive.- Since \( f^{(4)}(0) > 0 \), according to the generalization, \( f(0) = 0 \) is a local minimum.

Key Concepts

Local Extrema

When studying functions, it's important to understand where they reach their highest or lowest points within a particular range. Such points are called local extrema — local maxima or minima.
Local maxima occur where a function reaches a peak value before decreasing in value. Conversely, local minima are where the function is at its lowest point before increasing.
These points are not the absolute highs or lows, but are specific to a local region or interval of the function's graph.

Derivatives

Derivatives help us understand the rate at which a function changes at any given point. The first derivative of a function, given by \( f'(x) \), tells us about the slope or the rate of change of the function.
- If \( f'(x) > 0 \), the function is increasing.- If \( f'(x) < 0 \), it is decreasing.- If \( f'(x) = 0 \), the function might have a turning point, such as a local maximum or minimum.

Higher order derivatives, like the second derivative \( f''(x) \), offer deeper insight into the function's concavity and can help determine local extrema through tests like the Second Derivative Test.

Calculus Concepts

In calculus, the study of change and motion, understanding the behavior of functions at critical points is crucial. The derivatives play a vital role here.
The Second Derivative Test for finding local extrema states:- If \( f''(c) > 0 \), the function is concave up, indicating a local minimum.- If \( f''(c) < 0 \), the function is concave down, indicating a local maximum.

However, if \( f''(c) = 0 \), the test is inconclusive, necessitating higher-order tests or alternative approaches. That's where the concept of generalizing extensions of tests steps in to offer solutions, as described in the exercise with \( f(x) = x^4 \).

Local Minimum

A local minimum is a point where the function value is less than any other value in its vicinity.
For instance, in our exercise on \( f(x) = x^4 \), by calculating the derivatives and applying the generalized test, we find that \( f(0) = 0 \) is a local minimum.

Since all derivatives up to the third order are zero ( \( f'(0) = f''(0) = f'''(0) = 0 \) ) and the first non-zero derivative is positive, \( f^{(4)}(0) = 24 \), we conclude there's a local minimum at \( x = 0 \). This approach allows us to determine extrema when standard methods fall short.