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

Short answer

The exact \(x\)-intercepts, maximums, minimums, and increasing/decreasing intervals will depend on the solutions to \(g'(x) = 0\) and the sign of \(g''(x)\).

Step-by-step solution

Graphing the Function

Firstly, graph the function \( g(x) = x^4 - 5x^3 + 2x^2 + x - 3 \). The shape of the graph is dictated by the leading coefficient, \(x^4\), which tells us that both ends of the graph point upwards.

Finding the \(x\)-Intercepts

The \(x\)-intercepts are the values of \(x\) where the function equals zero. Therefore, we solve the equation \(g(x) = 0\) or \(x^4 - 5x^3 + 2x^2 + x - 3 = 0\). This may require numerical methods or software to solve, if it does not factor neatly.

Identifying Maximums and Minimums

To find the local maximums and minimums, we use calculus. The maxima and minima occur where the derivative of the function equals zero, and the value of the second derivative changes sign. Therefore, find the derivative of \(g(x)\) and set it equal to zero, solving for \(x\). These points are potential maximums and minimums - verify this using the second derivative and checking if it changes sign.

Determine Increasing and Decreasing Intervals

The function is increasing where its derivative is positive and decreasing where it's negative. To determine these intervals, calculate the derivative of \(g(x)\), which was found in step 3, and determine the intervals in which this derivative is positive or negative, respectively.

Key concepts

Understanding x-Intercepts of Functions

The x-intercepts of a function are the points where the function crosses the x-axis. In other words, these are the values of x for which the function output, f(x), is zero. To find the x-intercepts algebraically, we set the function equal to zero and solve for x. Graphically, you can see these points where the curve meets the horizontal axis.

For the polynomial function given by g(x) = x^4 - 5x^3 + 2x^2 + x - 3, we look for solutions to the equation g(x) = 0. Solving such a high-degree polynomial equation might be complex and typically requires using numerical methods or graphing software if the polynomial doesn't factor easily. The x-intercepts are crucial for graphing the function and understanding its behavior as they mark the transition of the curve from above the x-axis to below it, or vice versa.

Identifying Local Maximums and Minimums

Local maximums and minimums, often called extremas, represent the peaks and valleys of a function's graph. A local maximum is a point where the function changes from increasing to decreasing. Conversely, a local minimum is where it changes from decreasing to increasing. To identify these points, we use calculus, specifically the first and second derivatives of the function.

Start by finding the first derivative, g'(x), and set it equal to zero to find critical points. These are the candidates for local maximums and minimums. However, not all critical points are extremas, so we verify by using the second derivative, g''(x), or by analyzing the signs of the first derivative before and after each critical point. A sign change from positive to negative in the first derivative indicates a local maximum, while a sign change from negative to positive suggests a local minimum. The second derivative test can also provide insights: if g''(x) is positive at a critical point, it's likely a local minimum, and if it's negative, it suggests a local maximum.

Determining Increasing and Decreasing Intervals

A function's increasing and decreasing intervals show us where the function is going up or down as we move along the x-axis. Specifically, the function is increasing on intervals where the derivative, g'(x), is positive, and it's decreasing where this derivative is negative.

Graphically, imagine walking from left to right along the function's graph. When walking uphill, you're on an increasing interval; when heading downhill, you're on a decreasing interval. To find these intervals algebraically for the given function g(x) = x^4 - 5x^3 + 2x^2 + x - 3, we compute g'(x) and examine the sign of this derivative across the domain of the function. By testing values within the intervals defined by the critical points, we can determine where g'(x) is positive (increasing) or negative (decreasing). This analysis helps sketch the behavior of the function and is vital in understanding the overall dynamics of polynomial functions.