Question

Use a graphing utility to graph the function, approximate the relative minimum or maximum of the function, and estimate the open intervals on which the function is increasing or decreasing. \(f(x)=\frac{1}{4}\left(-4 x^{4}-5 x^{3}+10 x^{2}+8 x+6\right)\)

Short answer

The graph of the function \(f(x)=\frac{1}{4}\left(-4 x^{4}-5 x^{3}+10 x^{2}+8 x+6\right)\) can be sketched using a graphing utility. By examining the graph, the relative minimum or maximum as well as the intervals where the function is increasing and decreasing can be approximated.

Step-by-step solution

Graph the function

Using a graphing utility, plot the given function. Identify and note the rough shape of the graph, where it peaks and valleys, and where it seems to be increasing or decreasing.

Identify relative minimum and maximum

Examine the graph for peaks and valleys, these are potential relative maximum and minimum points respectively. The relative maximum is the highest point or peak in a particular section of the graph. Conversely, the relative minimum is the lowest point or the valley in a particular section of the graph.

Determine the intervals of increase and decrease

The function is increasing where the function is going up as you move from left to right along the x-axis, and decreasing where it goes down. Observing the graph, identify these intervals.

Key concepts

Relative Minimum

A relative minimum of a function is a point where the function value is lower than any other nearby points. Imagine a valley in a hill landscape—the lowest point of this valley is the relative minimum. In practice, a relative minimum can be spotted on a graph where the curve dips to its lowest local value within a small neighborhood. In mathematical terms, if a function \( f(x) \) has a relative minimum at \( x = c \), then \( f(c) \) is less than all nearby \( f(x) \) values within a certain range of \( c \). To find these points, use calculus techniques involving the first and second derivatives or, for broader learning, use graph approximations through devices like graphing calculators. A helpful sign to locate a relative minimum is when the slope of the function transitions from negative to positive, indicating a shift from decreasing to increasing angles on the graph.

Relative Maximum

A relative maximum is the opposite of a relative minimum. It is a point where the function reaches its highest local value compared to surrounding points. Using our valley and hill analogy, it is like the peak of a hill in the landscape. On a graph, you can spot relative maxima by looking for the highest point in any small area of the graph. For a function \( f(x) \), if there is a relative maximum at \( x = c \), then \( f(c) \) will be greater than all \( f(x) \) values surrounding \( c \) within a specific neighborhood range. To determine a relative maximum point, examine where the slope of the function changes from positive to negative, suggesting a shift from increasing to decreasing angles. This involves using calculus terms, detecting derivative zero points while observing changes from increasing to decreasing trends on the graph.

Intervals of Increase

Intervals of increase are where the function is rising as you move rightward along the x-axis. Imagine driving up an ascending road; these stretches are the function's increasing intervals. Mathematically, on an interval \( I \) where a function \( f(x) \) is increasing, for any two points \( a \) and \( b \) in \( I \), if \( a < b \) then \( f(a) < f(b) \). A function is increasing where the derivative \( f'(x) > 0 \). By observing a graph, you'll notice the curve going upward in these intervals. Identifying intervals of increase provides insight into the behavior of function growth across its domain which is essential for understanding its overall trend.

Intervals of Decrease

Intervals of decrease denote sections where the function heads downward as you go from left to right on the x-axis. This is akin to traveling downhill on a steep road. In these intervals, for any two points \( a \) and \( b \), if \( a < b \), then \( f(a) > f(b) \), emphasizing reductions in function values. Look at the graph to verify intervals of decrease where the graph swoops downward. Mathematically, these intervals occur where the derivative \( f'(x) < 0 \). Recognizing these declining sections helps determine where a function's value is contracting, a critical aspect in studying its complete behavior. Understanding intervals of decrease is essential for predicting where declines happen, aiding in ensuring accurate comprehensive analysis of the function's characteristics.