Question

In Exercise, use a graphing utility to estimate graphically all relative extrema of the function. $$ f(x)=5+3 x^{2}-x^{3} $$

Short answer

The short answer depends on the exact position of the extrema on the graph, which can vary slightly depending on the graphing utility used. The relative extrema can be estimated through the graph.

Step-by-step solution

Graph the Function

Use a graphing utility to plot the function \( f(x)=5+3 x^{2}-x^{3} \).

Identify Relative Extrema

Look for the highest and lowest points on the graph. These points are the relative maximum and minimum, respectively. They occur where the function changes direction, that is, where the slope of the function (derivative) changes its sign.

Estimate Extrema

Estimate the x-coordinates of these extrema by locating the points on the x-axis directly below or above the extrema points on the curve.

Key concepts

Graphing Utility

A graphing utility is an indispensable tool in calculus, especially when dealing with complex functions. It provides a visual representation of a function's behavior, which aids in better understanding the mathematical concepts. A graphing utility can be a dedicated graphing calculator, specialized software, or online applications that allow users to input a function and see its graph.

When plotting the function \( f(x)=5+3x^2-x^3 \), a graphing utility helps to identify important features such as intercepts, symmetry, and most importantly, areas where the function reaches its highest and lowest values, known as relative extrema. The visual aid from a graph can make it easier to understand the general shape and the behavior of the function – whether it's increasing, decreasing or turning – and serves to complement algebraic methods. For estimations and approximate solutions, a good graphing can be just what a student needs to bridge the gap between concept and application.

Relative Maximum and Minimum

The concepts of relative maximum and minimum are crucial in understanding how a function behaves. They refer to the highest and lowest points in a certain interval of a function's domain, respectively. These points represent 'peaks' and 'valleys' on the curve of the graph.

To locate a relative maximum, look for the points on the graph where the curve stops ascending and begins to descend. Similarly, a relative minimum is found where the curve stops descending and starts to ascend. These points are pivotal in many real-world applications such as finding the maximum profit or the minimum cost. It's important to note that a function may have several relative maxima and minima, and using a graphing utility to approximate these points can give a clear and immediate understanding of the function's extrema and overall shape.

Derivative Sign Change

The derivative of a function gives us information about the function's slope at any given point. When the sign of the derivative changes, it indicates a change in the direction of the curve – the transition from increasing to decreasing or vice versa. At a relative maximum, the derivative changes from positive to negative, indicating a hilltop on the graph. Conversely, at a relative minimum, the derivative changes from negative to positive, indicating a valley.

This sign change is a powerful tool in finding relative extrema. One can envision drawing a tangent line at various points on the curve: when the tangent line changes from sloping upwards to sloping downwards, a relative maximum is found, and when it changes from sloping downwards to sloping upwards, we find a relative minimum. Recognizing these derivative sign changes, especially with the help of a graphing tool, makes the identification of relative extrema relatively straightforward, which is a fundamental skill in calculus.