Problem 80 - Thomas Calculus

Question

Horizontal tangents True, or false? Explain. a. The graph of every polynomial of even degree (largest exponent even) has at least one horizontal tangent. b. The graph of every polynomial of odd degree (largest exponent odd) has at least one horizontal tangent. Parabolas.

Step-by-Step Solution

Answer

a. True b. False

Step 1: Identify the Problem

We need to determine whether the statements about polynomials having horizontal tangents are true or false. A horizontal tangent occurs where the derivative equals zero.

Step 2: Evaluate Polynomial of Even Degree

For part a, consider a polynomial of even degree, such as a quadratic function (degree 2). These functions can be expanded to have local maxima or minima. The derivative of such a polynomial will generally yield a valid solution where the derivative equals zero, indicating a horizontal tangent.

Step 3: Conclusion for Even Degree Polynomials

Since the derivative of a polynomial with even degree (e.g., a quadratic function) can easily be zero, such polynomials can and do have at least one horizontal tangent. Therefore, part a is true.

Step 4: Evaluate Polynomial of Odd Degree

For part b, consider a polynomial of odd degree. While the derivative can be zero, the nature of the even/odd degree indicates the end behavior (different at each end of the graph). This doesn't guarantee a horizontal tangent, as the graph could have a point of inflection without a local extremum causing a horizontal tangent.

Step 5: Conclusion for Odd Degree Polynomials

Polynomials of odd degree may not always have horizontal tangents simply due to the polynomial's structure. Therefore, part b is false.

Key Concepts

Horizontal Tangent

A horizontal tangent occurs on the graph of a function where the graph momentarily becomes flat. This means that at that point, the slope is zero. In mathematical terms, this is where the derivative of the function equals zero.

• Horizontal tangents are critical points that indicate either a local maximum, minimum, or a point of inflection depending on the behavior of the graph before and after the point.
• To find horizontal tangents, you calculate the derivative of the function and solve for where this derivative is equal to zero.

For example, if you have a function \( f(x) \), its derivative \( f'(x) \) will tell you the slope at any point \( x \). Setting \( f'(x) = 0 \) gives you the x-values where the horizontal tangents occur.

Even Degree Polynomial

An even degree polynomial has the highest power of its variable as an even number. Common examples include quadratic functions, like \( f(x) = ax^2 + bx + c \), where the highest power of \( x \) is 2.

• Even degree polynomials often have a U-shaped or inverted U-shaped graph, which guarantees that at least one horizontal tangent exists in the form of a local maximum or minimum.
• Since the graph can potentially rise and fall, depending on the leading coefficient, they will have turning points where the derivative equals zero.

These properties guarantee that any even degree polynomial will have one or more horizontal tangents because the graph will necessarily have points where it changes direction.

Odd Degree Polynomial

An odd degree polynomial, in contrast, has its highest power as an odd number, such as \( f(x) = ax^3 + bx^2 + cx + d \). These functions typically have graphs that extend from one quadrant to another.

• The graph of an odd degree polynomial generally crosses the x-axis, moving from negative to positive or vice versa, leading to no guarantee of a flat part (horizontal tangent).
• While the derivative \( f'(x) \) can still be zero, this does not ensure a local extremum like in even degree polynomials, as it might indicate a point of inflection.

The behavior at the ends of the graph (as \( x \) approaches \( +\infty \) or \( -\infty \)) is different, making it possible for an odd degree polynomial to have no points where the graph is flat.

Derivative

The derivative of a function provides a powerful insight into the behavior of its graph. Specifically, it tells us the rate of change or the slope of the graph at any point.

• When the derivative of a function equals zero, the graph has a horizontal tangent at that point.
• Derivatives are calculated using rules of differentiation, such as the power rule, the product rule, and the chain rule.

For example, if you have a polynomial \( f(x) = ax^n \), its derivative is \( f'(x) = nax^{n-1} \). To explore horizontal tangents, solve \( f'(x) = 0 \) to find the x-coordinates where the graph flattens out. Understanding derivatives is thus key to analyzing and predicting the graph behaviors of polynomial functions.