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. $$f(x)=x^5-4 x^3+x^2+2$$

Short answer

The graph is sketched. The x-intercepts are determined by setting the function equal to zero and solving for x. The local maximums and minimums are calculated by finding the derivative, setting it equal to zero, and solving for x. The intervals of increase and decrease are found by using the derivative and critical points.

Step-by-step solution

Graph the Function

Use a graphing tool to sketch the function \(f(x) = x^5 - 4x^3 + x^2 + 2\).

Identify the x-intercepts

The \(x\)-intercepts of the function are the values of \(x\) such that \(f(x) = 0\). Solve the equation \(x^5 - 4x^3 + x^2 + 2 = 0\) to find the \(x\)-intercepts.

Identify the Local Maximums and Minimums

To find the local maximums and minimums of the function, take the derivative of \(f(x)\), set it equal to zero, and solve for \(x\). Since it is a polynomial function of a higher degree, utilize techniques such as the quadratic formula or factoring to solve the derivative equation.

Determine Intervals of Increase and Decrease

Once the derivative of the function is found, the values of \(x\) where the derivative changes sign are critical points. For any interval between the critical points, pick a test point to determine whether the function is increasing or decreasing.

Key concepts

Graphing Functions

Graphing functions is an essential skill that helps you visualize mathematical relationships. For the given function, \(f(x) = x^5 - 4x^3 + x^2 + 2\), graphing provides insight into its behavior across different values of \(x\). Start by plotting the basic shape of the polynomial, which can be done using software tools or plotting points manually.

Since it's a fifth-degree polynomial, expect that it will have a general "W" shape, with at most four turning points. The "ends" of the graph, known as the end behavior, will approach \(\infty\) as \(x\) approaches \(\pm\infty\).

Graphing allows you to see where the curve crosses the \(x\)-axis, where peaks and valleys (local maxima and minima) occur, and how the function changes slope (increases or decreases). It's the first step in deeply understanding the characteristics of the function.

x-intercepts

The \(x\)-intercepts of a function occur where the graph crosses the \(x\)-axis. For the function \(f(x) = x^5 - 4x^3 + x^2 + 2\), these can be found by solving the equation \(f(x) = 0\). Finding \(x\)-intercepts for higher-degree polynomials requires solving the equation, which might involve complex numbers or numerical approximation if exact analytical solutions are difficult to find.

Here are some tips for finding \(x\)-intercepts:

• Factorize the polynomial, if possible.
• Use the Rational Root Theorem to test potential rational roots.
• Employ numerical methods or graphing tools to approximate roots if necessary.

Once identified, these \(x\)-intercepts are critical as they indicate real solutions where the function equals zero.

Local Extrema

Local extrema refer to the points on the graph where the function reaches local maximums or minimums. For the given polynomial function, these points are located by finding the derivative of the function and setting it to zero. This process identifies critical points, potential locations for local peaks or valleys.

To find local maxima and minima:

• Differentiate the function to obtain \(f'(x)\).
• Set \(f'(x) = 0\) and solve for \(x\).
• Analyze the sign change of \(f'(x)\) around these \(x\) values to determine if they are indeed local maxima or minima.

Such critical points are valuable as they provide insight into the behavior and shape of the function.

Increasing and Decreasing Intervals

Identifying intervals where a function is increasing or decreasing involves analyzing the derivative of the function. When \(f'(x) > 0\), the function is increasing; when \(f'(x) < 0\), it is decreasing. These intervals tell you about the general trend of the function's graph.

To determine these intervals:

• Use the critical points found from setting \(f'(x) = 0\).
• Choose a test point within each interval between the critical points.
• Substitute into \(f'(x)\) to check sign positivity or negativity.

Understanding increasing and decreasing behavior not only aids in graphing but also in predicting function behavior without the need for a complete graph, which is crucial in a range of applications.