Problem 39 - College Algebra with Corequisite Support

Question

For the following exercises, use a graphing utility to estimate the local extrema of each function and to estimate the intervals on which the function is increasing and decreasing. \(m(x)=x^{4}+2 x^{3}-12 x^{2}-10 x+4\)

Step-by-Step Solution

Answer

The function has local extrema at approximately (x-values), it increases on specific intervals and decreases on others, based on the derivative sign.

Step 1: Graph the Function

Use a graphing utility like Desmos or a graphing calculator to plot the function \(m(x)=x^{4}+2x^{3}-12x^{2}-10x+4\). Make sure to clearly see the shape of the curve by adjusting the window to capture all critical points where the graph changes direction.

Step 2: Identify Local Extrema

From the graph, look for the points where the function appears to reach local maxima or minima. These are the peaks and valleys of the graph. Take note of the approximate x-coordinates where these occur.

Step 3: Verify Extrema with Derivative

Calculate the derivative of the function: \(m'(x)=4x^3+6x^2-24x-10\). Set \(m'(x) = 0\) to find critical points. This gives you the x-values where local extrema might occur. Use these values to further check and validate the extrema found using the graph.

Step 4: Determine Increasing Intervals

Examine the graph from left to right and identify intervals where the function is going upwards. Alternatively, analyze the sign of the derivative, \(m'(x)\), to determine intervals where \(m'(x)>0\). This indicates the function is increasing.

Step 5: Determine Decreasing Intervals

Likewise, identify intervals on the graph where the function descends. This can also be done by finding where the derivative \(m'(x)<0\). These intervals indicate where the function is decreasing.

Key Concepts

Local Extrema

In mathematics, local extrema are the points on a function's graph where there is either a peak (local maximum) or a valley (local minimum). These points are crucial because they show the function's behavior in small regions. For example, in the function \(m(x)=x^{4}+2x^{3}-12x^{2}-10x+4\), finding these points visually through graphing helps in understanding and solving real-world problems that can be modeled by such polynomial functions.

Local extrema occur where the slope of the tangent to the curve is zero, meaning the derivative at those points is zero. This happens at the critical points. For our function, after plotting its graph, you can easily spot where it peaks or dips. If you lack a graphing utility, remember these essentials:

• Locating peaks (local maxima) and valleys (local minima) involves looking for where the graph changes direction.
• Math tools like graphing calculators or software can provide a clear visual representation.

Increasing and Decreasing Intervals

To determine the behavior of a function—when it goes up or down—you look at increasing and decreasing intervals. When graphing the polynomial \(m(x)=x^{4}+2x^{3}-12x^{2}-10x+4\), these intervals become apparent as sections where the curve ascends or descends.

Understanding these intervals not only helps in sketching accurate graphs but also in applying this knowledge practically.

• An increasing interval occurs when the function rises as \(x\) increases. We can see this when the graph trends upwards as you move left to right.
• A decreasing interval is where the function falls as \(x\) increases, evident from a downward slope in the graph.

To verify graphically estimated intervals, look at the derivative's sign. When \(m'(x)>0\), the function is increasing, and when \(m'(x)<0\), it’s decreasing. This analytical approach using derivatives can confirm what's observed visually.

Derivative Calculation

The derivative of a function is a powerful tool in calculus and essential for understanding how functions behave. It provides information on rates of change, particularly how fast or slow a function's value is changing at any given point.

For the function \(m(x)=x^{4}+2x^{3}-12x^{2}-10x+4\), its derivative \(m'(x)=4x^3+6x^2-24x-10\) helps to find critical points and thus, determine local extrema, increasing, and decreasing intervals.

• By setting \(m'(x)=0\), you find the critical points which potentially point to local maxima or minima.
• The derivative's sign also tells where the function is increasing or decreasing.

Succeeding in calculus often hinges on mastering derivative calculations since they unlock insights about the function’s behavior without always needing to graph it. Being familiar with differentiation rules and practicing consistently eases this learning curve.