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. $$h(x)=x^4-3 x^2+x$$

Short answer

The x-intercepts are \(x = 0, x = \pm \sqrt{3}\). There are local maximum at \(x = -1\) and a local minimum at \(x = 1\). The function increases on the intervals \((- \infty, -1)\) and \((1, \infty)\) and decreases on the interval \((-1, 1)\).

Step-by-step solution

1. Find the x-intercepts

To find the x-intercepts, set the function equal to zero and solve for x. This gives \[h(x) = x^4 - 3x^2 + x = 0\] Solving this equation gives the roots of the equation, which are the x-intercepts.

2. Identify local minimums and maximums

To find the local minimums and maximums, must first take the derivative of h(x): \[h'(x) = 4x^3 - 6x + 1\] Set the derivative equal to zero and solve for x, this will give the critical points. The second derivative test can then be applied to these critical points to determine whether they're local minimums or maximums. If the second derivative is positive at a point, it is a local minimum. If it's negative, the point is a local maximum.

3. Determine intervals of increase/decrease

Once the critical points from are found, these can be used to slice the x-axis into intervals. To determine if the function is increasing or decreasing over that interval, pick any x value in an interval then evaluate the derivative at that point. If it's positive then the function is increasing over that interval, if the derivative is negative then the function is decreasing.

4. Graph the function

The graph should include the x-intercepts, and local maximums/minimums. The regions of increase/decrease, taken from step 3, should also be clearly marked.

Key concepts

x-intercepts

When graphing polynomial functions, one important aspect to consider are the x-intercepts of the graph. An x-intercept is where the graph crosses the x-axis. This occurs where the function value is zero. For the function \(h(x) = x^4 - 3x^2 + x\), we can find the x-intercepts by setting the function equal to zero:

• \[x^4 - 3x^2 + x = 0\]

To solve this equation, factor out the common term \(x\), giving:

• \[x(x^3 - 3x + 1) = 0\]

The solutions to this equation, \(x = 0\) and the roots of the cubic \(x^3 - 3x + 1 = 0\), provide the x-intercepts. Understanding x-intercepts helps in sketching the overall behavior of the graph and where it crosses the x-axis.

local maximums and minimums

Finding local maximums and minimums is essential to understand a function's highest and lowest points within a given interval. These points help visualize the peaks and valleys on the graph. To find local maximums and minimums for the function \(h(x)\), we first take the derivative:

• \[h'(x) = 4x^3 - 6x + 1\]

Now, set the derivative equal to zero to locate critical points where possible maxima or minima occur:

• \[4x^3 - 6x + 1 = 0\]

Solving this will provide the critical points. To determine whether each critical point is a maximum or a minimum, use the second derivative test:

• If \(h''(x) > 0\), the point is a local minimum.
• If \(h''(x) < 0\), the point is a local maximum.

This analysis helps in understanding the contour of the polynomial function's graph.

derivatives

Derivatives are powerful tools in calculus used to analyze a function's behavior by providing the rate of change at any point. For polynomial functions, derivatives help identify changes in the direction of the graph. Specifically, the first derivative can indicate the slope (or rate of change) of the function. For our function, the first derivative \(h'(x) = 4x^3 - 6x + 1\) is key to finding critical points like local maximums and minimums, and the intervals where the graph increases or decreases.
Applying the derivative involves setting \(h'(x)\) to zero and solving for \(x\) to find these critical points. To further analyze these points, one can use the second derivative \(h''(x)\), which describes how the first derivative itself changes. This analysis is crucial for determining a local maximum or minimum and is essential when sketching the complete behavior of the graph.

intervals of increase and decrease

Identifying the intervals where a function increases or decreases requires understanding the function's behavior between critical points. Once the critical points are located by setting the first derivative \(h'(x) = 0\), use these points to divide the x-axis into intervals. Choose a test point from each interval and substitute it back into the first derivative:

• If \(h'(x) > 0\): The function is increasing on that interval.
• If \(h'(x) < 0\): The function is decreasing on that interval.

This approach systematically traces how the graph behaves in different regions, detecting slopes trending upwards (increasing) or downwards (decreasing). Recognizing these intervals gives important insights into the overall shape and direction of the polynomial graph, allowing you to clearly graph each segment.