Problem 33 - Calculus Single Variable

Question

Find each critical point \(c\) of the given function \(f\). Then use the First Derivative Test to determine whether \(f(c)\) is a local maximum value, a local minimum value, or neither. $$ f(x)=3 x-x^{1 / 3} $$

Step-by-Step Solution

Answer

The critical points are \( x = \frac{1}{27} \) and \( x = 0 \). \( x = \frac{1}{27} \) is a local minimum, \( x = 0 \) is neither.

Step 1: Find the First Derivative

To find the critical points, we first need to compute the first derivative of the function. Given the function \( f(x) = 3x - x^{1/3} \), we'll use differentiation rules. The derivative of \(3x\) is 3, and the derivative of \(x^{1/3}\) is \(\frac{1}{3} x^{-2/3}\). Therefore, the first derivative \( f'(x) \) is \[ f'(x) = 3 - \frac{1}{3} x^{-2/3} = 3 - \frac{1}{3x^{2/3}}. \]

Step 2: Solve for Critical Points

Critical points occur where the first derivative is zero or undefined. Set \( f'(x) = 0 \) and solve for \( x \). \[ 3 - \frac{1}{3x^{2/3}} = 0 \] implies \[ 3 = \frac{1}{3x^{2/3}}, \] which means \[ 9x^{2/3} = 1. \] Solving for \( x \) gives \[ x^{2/3} = \frac{1}{9}, \] leading to \[ x = \left(\frac{1}{9}\right)^{3/2} = \frac{1}{27}. \] Also, \( x = 0 \) is a critical point because the derivative \( f'(x) \) is undefined there.

Step 3: Use the First Derivative Test

To classify the critical points, use the first derivative test. Examine the sign of \( f'(x) \) around \( x = \frac{1}{27} \) and \( x = 0 \).1. For \( x = \frac{1}{27} \): - Choose a point slightly less than \( \frac{1}{27} \), say \( x = \frac{1}{30} \). - \( f'\left(\frac{1}{30}\right) = 3 - 90 \), which is negative, indicating \( f \) is decreasing. - Choose a point slightly more than \( \frac{1}{27} \), say \( x = \frac{1}{20} \). - \( f'\left(\frac{1}{20}\right) = 3 - 30 \), which is positive, indicating \( f \) is increasing. - So, \( x = \frac{1}{27} \) is a local minimum.2. For \( x = 0 \): - Consider the behavior as \( x \to 0^- \) and \( x \to 0^+ \). - For \( x < 0 \), \( f'\left(x\right) < 0 \) and for \( x > 0 \), \( f'\left(x\right) > 0 \). - Hence, \( x = 0 \) is neither a maximum nor a minimum.

Key Concepts

First Derivative Test

The First Derivative Test is a handy tool in calculus to determine whether a critical point of a function is a local maximum, local minimum, or neither. It revolves around examining the behavior of the first derivative around the critical points. A critical point occurs when the first derivative, given by \( f'(x) \), is either zero or undefined.

Here's how it works:

• If the derivative changes from negative to positive, the function transitions from decreasing to increasing. This indicates a local minimum.
• If the derivative changes from positive to negative, the function moves from increasing to decreasing. This signals a local maximum.
• If there is no sign change, then the point is neither a maximum nor a minimum.

By applying this test to critical points, like \( x = \frac{1}{27} \) and \( x = 0 \) in the given function, we can confirm the local behavior of the function at those points.

Local Minimum

A local minimum is a point in a function where the function value is smaller than all nearby values, creating a valley-like point on the function graph. At a local minimum, the slope or the derivative of the function changes from negative to positive, meaning the function starts to ascend after descending.

In the provided function \( f(x) = 3x - x^{1/3} \), we found a critical point at \( x = \frac{1}{27} \) using the first derivative test:

• Before \( x = \frac{1}{27} \), \( f'(x) < 0 \), indicating the function is decreasing.
• After \( x = \frac{1}{27} \), \( f'(x) > 0 \), showing that the function is increasing.

Thus, \( x = \frac{1}{27} \) represents a local minimum of the function.

Differentiation

Differentiation is at the core of calculus, used to find the derivative of a function. This involves calculating how a function changes at any given point, providing insights into the function’s slope or rate of change.

For the function \( f(x) = 3x - x^{1/3} \), we performed differentiation to find \( f'(x) \). Using basic differentiation rules:

• The derivative of a linear term like \( 3x \) is 3.
• The derivative of \( x^{1/3} \) involves applying the power rule, resulting in \( \frac{1}{3}x^{-2/3} \).

Thus, the first derivative of the function is \( f'(x) = 3 - \frac{1}{3x^{2/3}} \). This derivative helps us determine critical points where the function’s behavior changes.

Function Analysis

Function analysis involves examining the behavior of functions to understand key features such as increasing/decreasing intervals, local maxima/minima, and critical points. It combines techniques like differentiation and the First Derivative Test to visualize how the function behaves across its domain.

In analyzing the function \( f(x) = 3x - x^{1/3} \), the steps are broken down as follows:

• Determine the first derivative \( f'(x) = 3 - \frac{1}{3x^{2/3}} \).
• Find critical points by solving \( f'(x) = 0 \) or where \( f'(x) \) is undefined.
• Use the First Derivative Test to identify local maxima, minima, or neither by checking changes in sign around the critical points.

This comprehensive analysis gives a clearer picture of how the function behaves at different points, helping to locate valleys and peaks in the curve.