Problem 41 - University Calculus: Early Transcendentals

Question

Find the function's absolute maximum and minimum values and say where they are assumed. $$f(x)=x^{4 / 3}, \quad-1 \leq x \leq 8$$

Step-by-Step Solution

Answer

The absolute minimum value is 0 at \( x = 0 \), and the absolute maximum value is 16 at \( x = 8 \).

Step 1: Find the derivative of the function

To find the critical points, we first need the derivative of the function. The function is given as \( f(x) = x^{4/3} \). Using the power rule, the derivative \( f'(x) \) is calculated as follows: \[ f'(x) = \frac{4}{3}x^{(4/3)-1} = \frac{4}{3}x^{1/3} \]

Step 2: Determine critical points

Critical points occur where the derivative is zero or undefined. The derivative is undefined when \( x^{1/3} \) doesn't exist, which happens when \( x = 0 \). Setting the derivative \( f'(x) = \frac{4}{3}x^{1/3} = 0 \), we find that there are no values of \( x \) where the derivative equals zero. Thus, the only critical point is at \( x = 0 \).

Step 3: Evaluate the function at critical points and endpoints

We need to evaluate \( f(x) \) at the critical point and the endpoints of the interval \( x = -1 \) and \( x = 8 \):- At \( x = -1 \): \[ f(-1) = (-1)^{4/3} = \left((-1)^4\right)^{1/3} = 1^{1/3} = 1 \]- At \( x = 0 \): \[ f(0) = 0^{4/3} = 0 \]- At \( x = 8 \): \[ f(8) = 8^{4/3} = (2^3)^{4/3} = 2^4 = 16 \]

Step 4: Identify the absolute maximum and minimum

Compare the function values at the critical point and endpoints: 1. \( f(-1) = 1 \)2. \( f(0) = 0 \)3. \( f(8) = 16 \)The absolute minimum value is \( f(0) = 0 \) at \( x = 0 \), and the absolute maximum value is \( f(8) = 16 \) at \( x = 8 \).

Key Concepts

Understanding the Derivative

In calculus, the derivative is a way to understand the behavior of a function. It provides us with the slope of the function at any given point. This is crucial for solving various mathematical problems, such as finding the rate of change or optimizing functions.
For the function given as \( f(x) = x^{4/3} \), the derivative, denoted as \( f'(x) \), helps us find critical points. By applying the power rule to calculate \( f'(x) \), we determine how the function increases or decreases.

• In essence, calculating a derivative means finding how a small change in \( x \) affects the value of \( f(x) \).
• It's represented mathematically as \( f'(x) \).

Understanding how to find the derivative lays the foundation for determining other critical aspects of functions, like critical points.

Critical Points and What They Mean

Critical points are where the behavior of a function changes, either by reaching a peak or a bottom or perhaps by switching its pace of increase or decrease. These points occur where the derivative of the function equals zero or is undefined.
In our problem, the only critical point we found was at \( x = 0 \), where the derivative \( \frac{4}{3}x^{1/3} \) becomes undefined.

• When \( f'(x) = 0 \), the function could be at a local maximum, local minimum, or neither.
• When \( f'(x) \) is undefined, it may also indicate a crucial change in behavior.

Identifying these critical points allows us to further evaluate the function's features and behavior across the defined interval of interest.

Finding Absolute Maximum and Minimum

An absolute maximum or minimum refers to the highest or lowest value a function can take within a specific interval. For this, you need to evaluate the function at the critical points and endpoints of the interval.
In the problem, the interval given is \( -1 \leq x \leq 8 \). By evaluating \( f(x) \) at \( x = -1 \), \( x = 0 \), and \( x = 8 \), we determined that:

• The absolute minimum is \( f(0) = 0 \).
• The absolute maximum is \( f(8) = 16 \).

Surveying both the critical points and the endpoints ensures no potential maximum or minimum is overlooked within the specified range.

Using the Power Rule

The power rule is a fundamental strategy in calculus used to calculate the derivative of functions with power expressions. It simplifies the process by breaking it into a straightforward application.
According to the power rule, for any function \( f(x) = x^n \), the derivative \( f'(x) \) is \( nx^{n-1} \).

For the function \( f(x) = x^{4/3} \), applying the power rule gives us the derivative \( f'(x) = \frac{4}{3}x^{1/3} \).

• All you need to do is multiply by the exponent \( n \) and reduce the power by one.
• This makes the rule especially handy for functions involving polynomial expressions.

Mastering the power rule reduces the complexity of finding derivatives and is essential for tackling calculus problems efficiently.