Problem 30 - Algebra and Trigonometry Real Mathematics, Real People

Question

(a) use a graphing utility to graph the function and (b) determine the open intervals on which the function is increasing, decreasing, or constant. $$f(x)=-x^{3 / 4}$$

Step-by-Step Solution

Answer

The function \(f(x)=-x^{3 / 4}\) decreases on the interval \([0, +\infty)\). The function is not defined or constant at any interval.

Step 1: Graphing the function

Use the graphing utility to plot the function \(f(x)=-x^{3 / 4}\). This function has a unique shape due to the negative sign and the fractional exponent.

Step 2: Finding the derivative of the function

Differentiate the function \(f(x)=-x^{3 / 4}\) to get the first derivative \[f'(x)= -\frac{3}{4}x^{-1 / 4}\].

Step 3: Determining the intervals of increase and decrease

The function increases where the derivative \(f'(x)\) is greater than 0 and decreases where \(f'(x)\) is less than 0. As the function contains a negative sign and a fractional power, the derivative is always negative, meaning that the function is always decreasing. However, because \(x\) cannot be negative (due to the positive fractional exponent), the function is only defined for \(x\geq 0\)

Key Concepts

Derivative

In calculus, the derivative of a function gives us valuable information about the rate at which the function's value is changing at any given point. When we have a function such as \(f(x) = -x^{3/4}\), the process of differentiation helps us find the derivative, \(f'(x)\). Differentiation involves taking the power rule, which states that the derivative of \(x^n\) is \(nx^{n-1}\). In our example, applying this rule results in \(f'(x) = -\frac{3}{4}x^{-1/4}\).

This derivative can tell us whether the function is increasing or decreasing at different points. By understanding the nature of \(f'(x)\), we can predict how the function behaves over different intervals. This is an important tool in math to understand how functions behave without needing a graphical representation. It is particularly useful in determining critical points and understanding the general trend of the function.

Increasing and Decreasing Intervals

When analyzing a function, one key aspect we look at is where the function is increasing or decreasing. A function is increasing on intervals where its derivative is positive, which means that as \(x\) increases, \(f(x)\) also increases. Conversely, a function decreases on intervals where its derivative is negative, indicating that as \(x\) increases, \(f(x)\) decreases.

For \(f(x) = -x^{3/4}\), we have found that \(f'(x) = -\frac{3}{4}x^{-1/4}\), which is negative for all values of \(x\) within the domain of the function (\(x \geq 0\)). Thus, the function is always decreasing wherever it is defined. There aren’t any intervals within its domain where it increases.

Intervals of Increase and Decrease

Identifying intervals of increase and decrease is essential for understanding the overall shape and behavior of a graph. For \(f(x) = -x^{3/4}\), because the derivative \(f'(x)\) is negative across the whole domain, the function does not have any increasing intervals in the traditional sense.

Instead, its nature is strictly decreasing for all permissible values of \(x\) (\(x \geq 0\)). Understanding these characteristics is crucial for sketching graphs by hand and interpreting them correctly. It also aids in predicting behavior at boundaries or specific critical values.

Fractional Exponents

Fractional exponents, like the one in \(f(x) = -x^{3/4}\), represent roots of numbers. For instance, \(x^{3/4}\) can be thought of as the fourth root of \(x\) cubed. Fractional exponents are useful as they offer a concise way to express complex roots, making algebraic manipulations simpler.

Working with fractional exponents can initially seem challenging, but with practice, it becomes intuitive. In our function, the presence of the negative sign in front of \(x^{3/4}\) flips the graph upside down, which introduces a set of unique properties, making the function decrease throughout its domain. Understanding fractional exponents is essential for interpreting and graphing such mathematical expressions accurately.