Question

\(f(x)=3 x^{4}-44 x^{3}+60 x^{2}\) (Hint: Two different graphing windows may be needed.)

Short answer

Question: Sketch the graph of the polynomial function \(f(x) = 3x^4 - 44x^3 + 60x^2\). Find the critical points, inflection points, intercepts, and intervals of increase/decrease and concavity. Use two different windows to analyze the function. Answer: The key points of the function are: - Critical points: (0,0), (2, 12), (5, -125). - Inflection points: (2, 12), \(\left(\frac{5}{3}, \frac{500}{27}\right)\). - x-intercepts: \(x = 0, 2, 5\). - y-intercept: (0, 0). Intervals of increase/decrease: - Increasing: \((-\infty, 0)\), \((2, 5)\) - Decreasing: \((0, 2)\), \((5, \infty)\) Intervals of concavity: - Concave down: \((-\infty, \frac{5}{3})\), \((2, \infty)\) - Concave up: \((\frac{5}{3}, 2)\) Window 1: \(x\in[-10, 10]\) and \(y\in[-10, 10]\). Window 2: \(x\in[-1, 6]\) and \(y\in[-30, 10]\).

Step-by-step solution

Calculate the first and second derivatives of the function

First, find the first derivative \(f'(x)\) to determine the critical points of the function. Then, find the second derivative \(f''(x)\) to determine the inflection points. \(f'(x) = 12x^3 - 132x^2 + 120x\) \(f''(x) = 36x^2 - 264x + 120\)

Find critical points and intervals of increase and decrease.

To find the critical points, set \(f'(x)\) to zero and solve for \(x\). \(12x^3 - 132x^2 + 120x = 0\) \(x(12x^2 - 132x + 120) = 0\) The critical points are when \(x=0\) or when \(12x^2 - 132x + 120 = 0\). Factoring the quadratic further: \(12(x^2 - 11x + 10) = 0\) Now, we can find the roots \(x_1\) and \(x_2\) by factoring or using the quadratic formula: \(x =\frac{-b \pm \sqrt{b^2-4ac}}{2a}\) \(x_1 = 2\) and \(x_2 = 5\) Now find Intervals of increase and decrease. Test a value in each interval determined by the critical points (\((-\infty, 0)\), \((0, 2)\), \((2, 5)\), and \((5, \infty)\)). For example, if we test \(x=-1\), we get \(f'(-1) > 0\), so \(f(x)\) is increasing in \((-\infty, 0)\). By analyzing the intervals, we have: - \(f(x)\) is increasing on \((-\infty, 0)\) - \(f(x)\) is decreasing on \((0, 2)\) - \(f(x)\) is increasing on \((2, 5)\) - \(f(x)\) is decreasing on \((5, \infty)\)

Find inflection points and intervals of concavity.

To find the inflection points, set \(f''(x)\) to zero and solve for \(x\). \(36x^2 - 264x + 120 = 0\) \(x_1 =\frac{-(-264) \pm \sqrt{(-264)^2-4\cdot36\cdot120}}{2\cdot36}\) \(x_1 = 2\) and \(x_2 = \frac{5}{3}\) Now find Intervals of concavity. Test a value in each interval determined by the inflection points (\((-\infty, \frac{5}{3})\), \((\frac{5}{3}, 2)\), and \((2, \infty)\)). For example, if we test \(x=1\), we get \(f''(1)<0\), so \(f(x)\) is concave down in \((-\infty, \frac{5}{3})\). By analyzing the intervals, we have: - \(f(x)\) is concave down on \((-\infty, \frac{5}{3})\) - \(f(x)\) is concave up on \((\frac{5}{3}, 2)\) - \(f(x)\) is concave down on \((2, \infty)\)

Find the intercepts of the function

To find the x-intercepts, set \(f(x)\) to zero and solve for \(x\). \(3x^4 - 44x^3 + 60x^2 = 0\) \(x^2(3x^2 - 44x + 60) = 0\) The x-intercepts are when \(x=0\) or when \(3x^2 - 44x + 60 = 0\). The roots are the same as in Step 2, which are \(x=2\) and \(x=5\). To find the y-intercept, plug in \(x = 0\): \(f(0) = 0\) The y-intercept is also \(0\).

Graph the function using two different windows

Now you have enough information to graph the function. Plot the points and curves from the information obtained in the previous steps. Window 1: Choose a standard window size, like \(x\in[-10, 10]\) and \(y\in[-10, 10]\). This window size will help you see the overall function, including the x-intercepts and y-intercept. Window 2: Choose a more focused window size, like \(x\in[-1, 6]\) and \(y\in[-30, 10]\). This window size helps you see more details, in particular the critical and inflection points, and the intervals of increase and decrease, as well as concavity. Remember that graphing calculators or software can also help you visualize the function. Make sure you plot the key points and analyze the function using the information from the previous steps.

Key concepts

First Derivative Test

Understanding the behavior of functions is a cornerstone of calculus, and the first derivative test provides a robust tool for analyzing the increasing and decreasing nature of functions. This test involves taking the first derivative (rate of change) of a function and using it to determine the critical points—where the derivative is zero or undefined. These points can indicate where the function might have a local maximum, a local minimum, or a horizontal inflection point.

When applying this to the function (f(x)=3 x^{4}-44 x^{3}+60 x^{2}, we set the first derivative (f'(x) equal to zero and solve for (x) to find the critical points. Once these points are located, we can test intervals around them to determine if the function is increasing or decreasing in that interval. For example, picking a number between two critical points and plugging it into the derivative will tell us the slope of the tangent at that point, and thus, whether the function is increasing (positive slope) or decreasing (negative slope) in that range. Through this test, we elucidated the key intervals of our function's behavior.

Second Derivative Test

While the first derivative sheds light on the monotonicity of a function, the second derivative test dives deeper into the curvature aspects, unveiling the function's concavity—whether it's concave up or down—and potential points of inflection. The second derivative gives us the rate at which the function's slope is changing.

In applying the second derivative test to our function, (f''(x)=36x^2 - 264x + 120) was set to zero to solve for possible inflection points. An inflection point is where the curvature changes, which can signal a transition between concave up (cup-shaped) and concave down (cap-shaped). By testing values in the intervals defined by these inflection points, it can be determined whether the function is concave up (positive second derivative) or concave down (negative second derivative). The function in question, given as (f(x)), reveals its concavity in different intervals when analyzed through the second derivative.

Concavity and Inflection Points

The curvature of a graph is a visual representation of how fast or slow a function is increasing or decreasing, and this is where concavity and inflection points become significant. Concave up graphs resemble a smile, where the slopes of the tangent lines are increasing. Conversely, concave down graphs resemble a frown, with decreasing slopes for tangent lines. Inflection points are where the graph changes concavity.

To identify the concavity of (f(x)=3x^{4}-44x^{3}+60x^{2}), we locate where the second derivative changes sign. These points are crucial as they are possible locations of inflection points. Examining our function's second derivative, we can say that in each interval outlined by its inflection points, the function exhibits distinct concavity for each section. Thus, we can sketch the general shape of the function's graph, a valuable asset in understanding the overall behavior of the function.

Graphing Polynomial Functions

Graphing is an essential step in visualizing and understanding the overall structure of polynomial functions. By plotting polynomial functions like (f(x)=3x^{4}-44x^{3}+60x^{2}), one can witness firsthand the rise and fall of the graph, the locations of maxima and minima, and the smooth, continuous nature of polynomial curves. The method involves plotting critical points, zeros (x-intercepts), and the y-intercept, then checking concavity and connecting these points with a smooth curve that reflects the increasing and decreasing intervals and curves up or down depending on concavity.

Using two different graphing windows as suggested, one with a larger view and another with a focused view, allows for a detailed analysis of finer graphical features, as well as an understanding of the broader behavior of the function. Importantly, by following the previous steps laid out in our derivative tests, we have substantial information to create a comprehensive and accurate graph of the polynomial function in question.