Problem 20 - Applied Calculus

Question

The function \(f(x)=x^{4}-4 x^{3}+8 x\) has a critical point at \(x=1 .\) Use the second-derivative test to identify it as a local maximum or local minimum.

Step-by-Step Solution

Answer

The critical point at \(x=1\) is a local maximum.

Step 1: Find First Derivative

To use the second-derivative test, first, we need to find the first derivative, \(f'(x)\), of the function \(f(x) = x^4 - 4x^3 + 8x\). Differentiate term by term: \[ f'(x) = 4x^3 - 12x^2 + 8. \]

Step 2: Verify Critical Point

We are provided with the critical point \(x=1\). Substitute \(x = 1\) into the first derivative \(f'(x)\) to verify that it is indeed a critical point:\[ f'(1) = 4(1)^3 - 12(1)^2 + 8 = 4 - 12 + 8 = 0. \]Since \(f'(1) = 0\), \(x=1\) is a critical point.

Step 3: Find Second Derivative

The second-derivative test requires us to find the second derivative, \(f''(x)\). Differentiate \(f'(x) = 4x^3 - 12x^2 + 8\) again:\[ f''(x) = 12x^2 - 24x.\]

Step 4: Evaluate Second Derivative at Critical Point

Substitute the critical point \(x = 1\) into the second derivative \(f''(x)\) to determine the nature of the critical point:\[ f''(1) = 12(1)^2 - 24(1) = 12 - 24 = -12. \]

Step 5: Determine Type of Critical Point

Utilize the second-derivative test, which states:- If \(f''(x) > 0\), the point is a local minimum.- If \(f''(x) < 0\), the point is a local maximum.Here, \(f''(1) = -12 \lt 0\), indicating that \(x = 1\) is a local maximum.

Key Concepts

Critical point

Critical points are crucial in calculus when trying to find where a function changes from increasing to decreasing (or vice versa), or to determine where the function's extremities (peaks and valleys) might be located. These are values of x where the first derivative, denoted by \( f'(x) \), is equal to zero or undefined. In simpler terms, at critical points, the slope of the tangent to the curve is zero, indicating a horizontal tangent line.
For example, in the function \( f(x) = x^4 - 4x^3 + 8x \), we identify \( x = 1 \) as a critical point because substituting into its first derivative results in \( f'(1) = 0 \). This confirms that the slope of the tangent line at this point is zero.
Recognizing and understanding critical points is a fundamental step in analyzing the behavior of functions and determining whether these points are locations of local minima, maxima, or saddle points.

Second-derivative test

The second-derivative test is a method used for determining the concavity of a function at its critical points, thus helping assess the nature of these points. After confirming a critical point by showing \( f'(x) = 0 \), we use the second derivative, \( f''(x) \), to decide if it is a local maximum, minimum, or neither.

• If \( f''(x) > 0 \) at the critical point, the function is concave up, suggesting a local minimum.
• If \( f''(x) < 0 \), the function is concave down, indicating a local maximum.
• If \( f''(x) = 0 \), the test is inconclusive, and there could be a possibility of an inflection point instead.

In our example, after finding the second derivative \( f''(x) = 12x^2 - 24x \), substituting \( x = 1 \) yields \( f''(1) = -12 \), which is less than zero. This clearly reveals that \( x = 1 \) is a local maximum.

First derivative

The first derivative of a function gives us information about the slope of the function or how the function changes with respect to x. In other words, it helps understand the rate at which the function rises or falls.
For our function \( f(x) = x^4 - 4x^3 + 8x \), the first derivative is calculated as \( f'(x) = 4x^3 - 12x^2 + 8 \). This derivative tells us where the function is increasing or decreasing based on its sign:

• If \( f'(x) > 0 \), the function is increasing at that point.
• If \( f'(x) < 0 \), the function is decreasing.

Finding zero points of \( f'(x) \) leads directly to identifying critical points where potential changes in behavior occur, as seen with the critical point at \( x = 1 \).
Understanding the first derivative gives a clear picture of the overall behavior and trends of the function.

Second derivative

The second derivative of a function is an integral tool for understanding the concavity and the rate at which the slope of the function is changing. It is essentially the derivative of the first derivative. While the first derivative tells us about the slope, the second derivative indicates if that slope is increasing or decreasing, thus showing concavity.
For example, in our case, we found the second derivative of \( f(x) = x^4 - 4x^3 + 8x \) to be \( f''(x) = 12x^2 - 24x \). By substituting the critical point into this, \( f''(1) = -12 \) gave a clear indication that the function is concave down at \( x = 1 \).

• Concave down indicates a local maximum.
• Concave up implies a local minimum.

Thus, the second derivative test not only confirms the extremities found with critical points but also helps in more precisely characterizing their nature.