Question

Graph the function. Identify the \(x\)-intercepts and the points where the local maximums and local minimums occur. Determine the intervals for which the function is increasing or decreasing. $$f(x)=0.5 x^3-2 x+2.5$$

Short answer

The graph of the function is a cubic graph. The x-intercepts, local maximums and minimums depend on the roots of the cubic equation and the sign of the derivative. The intervals of increasing and decreasing can be found by testing the sign of the derivative in different intervals.

Step-by-step solution

Graph the function

Graph the function \(f(x) = 0.5x^3 - 2x + 2.5\). This graph can be created by plotting points or generated by computational tools.

Find the x-intercepts

The x-intercepts are the points where the graph of the function crosses the x-axis. Mathematically, these are the points for which \(f(x) = 0\). Therefore, solve the equation \(0.5x^3 - 2x + 2.5 = 0\) for \(x\).

Find local maximums and minimums

The local maximums and minimums occur where the derivative of the function is zero or undefined. First, obtain the derivative \(f'(x)\) by differentiating \(f(x)\). Then solve \(f'(x) = 0\) for \(x\) to get the critical points. Determine whether each critical point is a local maximum, a local minimum, or neither by using the first or second derivative test.

Identify the intervals of increasing and decreasing

The function is increasing where its derivative is positive and decreasing where its derivative is negative. To find these intervals, sketch a number line labeling the critical points found in Step 3, then use test points in each interval to determine the sign of \(f'(x)\) in that interval.

Key concepts

X-Intercepts

Understanding the x-intercepts of a polynomial function is fundamental when analyzing its behavior. X-intercepts, also known as zeros or roots, are the points where the graph of a function crosses or touches the x-axis. To find the x-intercepts of the function f(x) = 0.5x^3 - 2x + 2.5, set f(x) equal to zero and solve for the values of x that satisfy the equation.

For the given function, solving 0.5x^3 - 2x + 2.5 = 0 might require numerical methods or the use of a graphing calculator since the equation does not factor easily. The x-intercepts are crucial for sketching the rough shape of the graph and serve as a starting point for further analysis of the function's behavior.

Local Maximums and Minimums

Local maximums and minimums, also referred to as extrema, represent the peaks and valleys of a function's graph. These points are where the function changes direction, moving from increasing to decreasing or vice versa. To find these critical points for the function f(x) = 0.5x^3 - 2x + 2.5, calculate the first derivative f'(x), then set it equal to zero to find potential extrema.

By differentiating, we obtain f'(x) = 1.5x^2 - 2. Solving 1.5x^2 - 2 = 0 gives us the values of x where the slope is zero; these are our candidates for local maxima and minima. After finding these points, use further tests to discern whether they are indeed maxima, minima, or points of inflection.

Intervals of Increase and Decrease

The intervals of increase and decrease are segments within the domain of a function where its values grow or diminish. Defining these intervals helps in understanding the overall trend of a function's graph. For the function f(x) = 0.5x^3 - 2x + 2.5, these intervals can be determined by analyzing the sign of the first derivative, f'(x).

By plotting the critical points on a number line and testing points between them, we can discern where f'(x) is positive or negative. If f'(x) is positive on an interval, the function is increasing there; conversely, if it's negative, the function is decreasing. This step provides valuable information for sketching the function's behavior and is essential when constructing a graph by hand or interpreting one made by software.

First and Second Derivative Test

The first and second derivative tests are powerful tools used to classify the critical points as local maximums, minimums, or points of inflection. After finding the critical points of f(x) by setting f'(x) to zero, use the first derivative test to check the sign of f'(x) before and after each critical point.

For a local maximum, f'(x) will change from positive to negative as you pass through the critical point; for a local minimum, it will change from negative to positive. If the first derivative does not change sign, the point might be an inflection point. The second derivative—f''(x), provides additional insight: if it's positive at a critical point, the point is likely a local minimum, and if negative, a local maximum. These tests are crucial for confirming the nature of extrema and for the full interpretation of a function's graph.