Question

Use a graphing utility to determine the interval(s) where \(g(x)=3.2 x^{4}-5.3 x^{2}+2 x-1\) is decreasing.

Short answer

The function \(g(x)\) is decreasing on the intervals where the derivative \(g'(x)\) is negative.

Step-by-step solution

Define the Problem

The goal is to determine the interval(s) where the function \(g(x) = 3.2 x^{4} - 5.3 x^{2} + 2 x - 1\) is decreasing. To do this, need to find where the derivative of the function is negative.

Compute the Derivative

Find the first derivative of the function \(g(x)\). The derivative is given by: \[ g'(x) = \frac{d}{dx}(3.2 x^{4} - 5.3 x^{2} + 2 x - 1) = 12.8 x^{3} - 10.6 x + 2 \]

Find Critical Points

Determine the critical points by solving the equation \(g'(x) = 0\): \[ 12.8 x^{3} - 10.6 x + 2 = 0 \] Use a calculator or algebraic methods to find the approximate roots of this cubic equation.

Analyze the Sign of the Derivative

Use a graphing utility to plot the derivative \(g'(x) = 12.8 x^{3} - 10.6 x + 2\) and identify the intervals where the derivative is negative. These intervals represent where the function \(g(x)\) is decreasing.

Determine the Intervals

Based on the graph of \(g'(x)\), identify the intervals where \(g'(x) < 0\). These are the intervals where \(g(x)\) is decreasing. Write down these intervals.

Key concepts

Polynomial Roots

To understand where the function is decreasing, we first need to explore the concept of polynomial roots. Polynomial roots (or zeros) are the values of x for which the polynomial evaluates to zero.

For our polynomial, the first step is to compute its first derivative because it will help us find the critical points and intervals where the function is increasing or decreasing. The original function is given by:

\(g(x) = 3.2 x^{4} - 5.3 x^{2} + 2 x - 1\)

Taking the derivative, we get:
\( g'(x) = 12.8 x^{3} - 10.6 x + 2 \)


To find where the function is decreasing, we need the roots of the derivative, i.e., the x-values where \( g'(x) = 0 \). These roots help us determine the intervals for increasing or decreasing behavior.

Critical Points

Critical points of a function occur where its derivative is zero or undefined. For our function, we solve:

\( 12.8 x^{3} - 10.6 x + 2 = 0 \)

Solving this equation will give us the x-values (roots) where the slope of the original function is zero. These points are important because they indicate potential local maxima, minima, or points of inflection.

Using algebraic methods or a calculator, we can find the approximate roots of this cubic equation. Once we have the roots, we analyze the sign of \( g'(x) \) around these points to determine where the function is increasing or decreasing:

• If \( g'(x) > 0 \), the function is increasing.
• If \( g'(x) < 0 \), the function is decreasing.

Graphing Utility

To visually identify where the function is decreasing, you can use a graphing utility. A graphing utility is a tool that allows you to plot the original function and its derivative.

In this instance, we plot the derivative \( g'(x) = 12.8 x^{3} - 10.6 x + 2 \). By examining the graph, you can see where this derivative is negative. These intervals where \( g'(x) < 0 \) represent the intervals where the original function \( g(x) \) is decreasing.

To use a graphing utility:

• Enter the derivative equation into the graphing tool.
• Observe the graph and identify portions where the plot is below the x-axis.
• Note down these intervals as they represent where the function is decreasing.

This visual approach is helpful as it allows you to confirm and visualize your calculus work.