Problem 28 - Thomas Calculus

Question

In Exercises \(9-28 :\) a. Find the intervals on which the function is increasing and decreasing. b. Then identify the function's local extreme values, if any, saying where they are taken on. c. Which, if any, of the extreme values are absolute? d. Support your findings with a graphing calculator or computer grapher. $$ k(x)=x^{2 / 3}\left(x^{2}-4\right) $$

Step-by-Step Solution

Step 1: Identify key features

Determine intercepts, asymptotes, critical points, and end behavior.

Step 2: Plot the graph

Plot key features and connect them to form the complete graph.

Key Concepts

Derivatives

Derivatives are a cornerstone concept in calculus, crucial for understanding how functions change. They provide the rate of change of a function with respect to one of its variables. In simple terms, the derivative tells us how steep a curve is at any given point. Derivatives are often symbolized by \( f'(x) \) or \( \frac{df}{dx} \), where \( f(x) \) is a function of \( x \). The derivative reflects the slope of the tangent line to the curve at any given point. This slope can be positive, negative, or zero, indicating whether the function is increasing, decreasing, or constant at that point.
To find the derivative of a product of two functions, we use the product rule. It's essential to grasp this rule since many complex functions are indeed products of simpler functions.

Increasing and Decreasing Intervals

To determine where a function is increasing or decreasing, we look at the sign of its derivative.

• If the derivative \( f'(x) > 0 \), the function is increasing on that interval.
• If \( f'(x) < 0 \), the function is decreasing on that interval.
• If \( f'(x) = 0 \), it may indicate a local maximum, minimum, or a saddle point. Hence, further analysis might be required.

In our given function \( k(x) \), we computed the derivative \( k'(x) \). The goal is to solve the inequality \( k'(x) > 0 \) to find where the function increases and \( k'(x) < 0 \) to find where it decreases. Testing critical points, where the derivative is zero or undefined, helps in segmenting the function into intervals for examination.

Local and Absolute Extrema

Extrema refer to the extreme values a function can take, such as maximums and minimums. Local extrema are extreme values within a limited region of the function, while absolute extrema are the highest or lowest points over the entire domain of the function.

• A local maximum is a point where the function reaches a peak in its neighborhood.
• A local minimum is a trough or the lowest point in its neighborhood.

To find these extremas, we look for places where the derivative \( f'(x) \) changes sign. For example, if \( f'(x) \) transitions from positive to negative, there's likely a local maximum. Meanwhile, a shift from negative to positive indicates a local minimum.
To identify absolute extrema among these local extrema, evaluate the function's value across its domain or any boundaries within the problem.

Product Rule

The product rule is a fundamental tool in derivatives, enabling us to differentiate expressions where two functions are multiplied together.
It is expressed as \( (f(x)g(x))' = f'(x)g(x) + f(x)g'(x) \). This rule states that the derivative of a product of two functions is the derivative of the first function times the second function plus the first function times the derivative of the second function.

• First, identify your functions as \( f(x) \) and \( g(x) \).
• Calculate the respective derivatives \( f'(x) \) and \( g'(x) \).
• Apply the rule as outlined to find the derivative of their product.

In our exercise, we used the product rule to find the derivative of the function \( k(x) = x^{2/3}(x^2 - 4) \), which was essential for further analysis.