Question

In Exercises, find the critical numbers and the open intervals on which the function is increasing or decreasing. Then use a graphing utility to graph the function. $$ f(x)=x^{4}-2 x^{3} $$

Short answer

The critical points of the function \(f(x) = x^4 - 2x^3\) are \(x = 0\) and \(x = \frac{3}{2}\). The function is decreasing on the interval \((- \infty, 0)\), and increasing on the intervals \((0, \frac{3}{2})\) and \((\frac{3}{2}, \infty)\).

Step-by-step solution

Calculating the derivative

To find the critical points, we need to first determine the derivative of the function. This can be achieved with the power rule, which states that the derivative of \(x^n\) is \(n*x^{(n-1)}\). Using this rule, the derivative \(f'(x)\) of our function \(f(x) = x^{4} - 2x^{3}\) is \(f'(x) = 4x^{3} - 6x^{2}\).

Derivative equals zero

To find the critical points, we must solve for \(x\) in the equation \(f'(x) = 0\). Rearranging \(4x^{3} - 6x^{2} = 0\) and factoring out \(2x^{2}\) gives us \(2x^{2}(2x-3) = 0\). This equation has two roots (i.e., \(x = 0\) and \(x = \frac{3}{2}\)), which are our critical points.

Determine Intervals of Increase and Decrease

From Step 2, we derived two critical points: \(x = 0\) and \(x = \frac{3}{2}\). For an interval test, we must select any point within these intervals: \((- \infty, 0)\), \((0, \frac{3}{2})\), and \((\frac{3}{2}, \infty)\). Plugging these points into \(f'(x)\) gives values of the derivative's sign, indicating whether the function is increasing or decreasing. For \(f'(x) < 0\), the function decreases, and for \(f'(x) > 0\), the function increases. In this case, the function is decreasing in the interval \((- \infty, 0)\) and increasing in the intervals \((0, \frac{3}{2})\) and \((\frac{3}{2}, \infty)\).

Graph the function

The last step is to plot the function to visually comprehend the derived properties above. Use any graphic calculator or software such as Desmos or Geogebra to plot \(f(x) = x^{4} - 2x^{3}\). It will clearly show how the function decreases before 0 and after \(\frac{3}{2}\) and increases between 0 and \(\frac{3}{2}\).

Key concepts

Derivative of Polynomial Functions

Understanding derivatives is fundamental in calculus, especially when dealing with polynomial functions. A derivative essentially represents the rate at which a function is changing at any given point, and can be visualized as the slope of the tangent line to the function's graph at that point.

For polynomial functions, the power rule simplifies the process of finding derivatives. If you have a term like \(a x^n\), its derivative will be \(n \times a x^{n-1}\). Applying this rule term by term to the given function \(f(x) = x^4 - 2x^3\) results in the derivative \(f'(x) = 4x^3 - 6x^2\).

Computing the derivative is the first step when looking to identify critical numbers of a function, which are the points where the function's graph changes direction or where a local maximum or minimum occurs. These points are located where the derivative is zero or undefined.

Intervals of Increase and Decrease

Once we have the derivative of a function, we can determine where the function is increasing or decreasing. A positive derivative implies that the function is rising, while a negative derivative suggests it is falling. To analyze this behavior, we split the domain of the function into intervals using critical numbers as boundaries.

In our case, we solve \(f'(x) = 4x^3 - 6x^2 = 0\) to find the critical numbers, which are \(x = 0\) and \(x = \frac{3}{2}\). These critical numbers divide the domain into intervals: \((-\infty, 0)\), \((0, \frac{3}{2})\), and \((\frac{3}{2}, \infty)\).

To determine if the function is increasing or decreasing on these intervals, we test a point from each interval in the derivative function. If the result is positive, the original function is increasing on that interval. If negative, it is decreasing. According to our exercise, the function \(f(x)\) is found to be decreasing on \((-\infty, 0)\) and increasing on \((0, \frac{3}{2})\) and \((\frac{3}{2}, \infty)\).

Graphing Polynomial Functions

Graphing polynomial functions can give a visual representation of the concepts of derivatives and intervals of increase and decrease. By plotting the function, we can see these concepts in action. With the function \(f(x) = x^4 - 2x^3\), using graphing software or a graphing calculator, we can visually confirm the intervals of increase and decrease found analytically.

The graph will show a declining slope from \(-\infty\) to \(0\) and an increasing slope from there up to \(\frac{3}{2}\), after which it continues to ascend. The points where the slope of the graph changes—from decreasing to increasing—are precisely the critical numbers \(x = 0\) and \(x = \frac{3}{2}\) that we calculated. These points are where the graph either reaches a peak or a trough, crucial for understanding the overall behavior of the function.

Graphing is not only a powerful tool for visual learning but also serves as a way to double-check the work done analytically in the previous steps. It reinforces the relationship between the algebraic manipulation of functions and their geometric representation.