Turning Points Of Polynomial H(x) = 3x^4 - 4x^2 + 1 True Or False

Hey guys! Let's dive into the fascinating world of polynomials and their turning points. Today, we're tackling the statement: "The polynomial h(x) = 3x⁴ - 4x² + 1 has at most 4 turning points." We're going to break this down, explore what turning points are, and ultimately determine whether this statement is true or false. So, buckle up and get ready for a mathematical adventure!

Before we jump into the specifics of our polynomial, let's solidify our understanding of turning points. Turning points, also known as local maxima or minima, are crucial features of a polynomial's graph. Think of them as the peaks and valleys of the curve. A turning point is a point on the graph where the function changes its direction – from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum). These points are where the slope of the tangent line to the graph is zero, which is a key concept in calculus. To find these turning points, we often use derivatives, but we can also understand them visually by looking at the graph of the polynomial.

Polynomial functions, by their very nature, can have a certain number of these turning points, and this number is directly related to the degree of the polynomial. The degree of a polynomial is the highest power of the variable in the expression. For instance, in our example, h(x) = 3x⁴ - 4x² + 1, the degree is 4. A polynomial of degree n can have at most n - 1 turning points. This is a fundamental rule that helps us predict the behavior of polynomial graphs. It's important to note that a polynomial can have fewer than n - 1 turning points, but it cannot have more. This principle stems from the relationship between the polynomial and its derivative, which is another polynomial with a degree one less than the original. The roots of the derivative correspond to the turning points of the original polynomial. The concept of turning points is critical in various applications, from optimization problems in engineering to modeling real-world phenomena in physics and economics. Understanding where a function reaches its maximum or minimum value is invaluable in these contexts. So, as we delve deeper into our example, keep this key relationship in mind: the number of turning points is intimately linked to the degree of the polynomial. This connection is not just a mathematical curiosity but a powerful tool for analyzing and predicting the behavior of functions.

Now, let's focus on our specific polynomial, h(x) = 3x⁴ - 4x² + 1. As we already noted, the degree of this polynomial is 4. This means, according to the rule we just discussed, that h(x) can have at most 4 - 1 = 3 turning points. It's crucial to recognize this upper limit. The statement we're evaluating claims that it has at most 4 turning points, which seems plausible at first glance, but we need to be precise. The degree of the polynomial dictates the maximum number of turning points, but it doesn't guarantee that the polynomial will have that many.

To further analyze h(x), we can use a few techniques. One approach is to find the derivative of the polynomial. The derivative, denoted as h'(x), gives us the slope of the tangent line at any point on the graph of h(x). The points where the derivative is equal to zero correspond to the turning points. So, let's find the derivative: h'(x) = 12x³ - 8x. Now, we need to solve the equation h'(x) = 0 to find the x-coordinates of the turning points. We have 12x³ - 8x = 0, which can be factored as 4x(3x² - 2) = 0. This gives us three solutions: x = 0, x = √(2/3), and x = -√(2/3). These three x-values correspond to the points where the graph of h(x) changes direction – our turning points! We've found three distinct solutions for h'(x) = 0, which confirms that h(x) has exactly three turning points. This is less than the 4 turning points suggested by the original statement, but it is within the maximum limit of 3 that we deduced from the degree of the polynomial. It's important to remember that the derivative method provides a precise way to identify turning points, and it aligns perfectly with the theoretical limit based on the polynomial's degree. By examining the derivative, we gain a deeper understanding of the polynomial's behavior and its critical points.

Another helpful way to understand turning points is to visualize the graph of the polynomial. You can use graphing software or online tools to plot h(x) = 3x⁴ - 4x² + 1. When you look at the graph, you'll clearly see the turning points – the local maxima and minima. The graph of h(x) will have a characteristic 'W' shape. This shape is typical for quartic (degree 4) polynomials with a positive leading coefficient (in our case, 3). The 'W' shape visually confirms the existence of three turning points: two local minima and one local maximum. The minima occur at the points where the graph dips down, and the maximum occurs where the graph peaks between the minima. Visualizing the graph complements our earlier analysis using the derivative. The graph provides an intuitive understanding of the turning points, while the derivative method gives us a precise way to calculate their locations. This combination of visual and analytical approaches offers a comprehensive picture of the polynomial's behavior. When examining the graph, pay attention to the symmetry as well. The polynomial h(x) is an even function because all the powers of x are even (4 and 2). This means that the graph is symmetric with respect to the y-axis. This symmetry is reflected in the turning points as well. The two minima are located at x = √(2/3) and x = -√(2/3), which are symmetric about the y-axis. The maximum is located at x = 0, right on the y-axis. The visual representation of the polynomial helps to solidify the concepts we've discussed and provides a valuable tool for understanding turning points and polynomial behavior in general.

So, let's revisit the original statement: "The polynomial h(x) = 3x⁴ - 4x² + 1 has at most 4 turning points." Based on our analysis, we know that a polynomial of degree 4 can have at most 3 turning points. We found the derivative of h(x), solved for its roots, and identified three distinct turning points. We also visualized the graph of h(x), which confirmed the presence of these three turning points. Therefore, while it's technically true that h(x) has at most 4 turning points (since 3 is less than 4), the statement is not the most accurate representation of the situation. The most accurate statement would be that h(x) has at most 3 turning points. However, focusing on the strict interpretation of