Problem 53 - Essential Calculus

Question

Determine whether the statement is true or false. If it is false, explain why or give an example that shows it is false. If the graph of a polynomial function has three \(x\) -intercepts, then it must have at least two points at which its tangent line is horizontal.

Step-by-Step Solution

Answer

The statement is false. For instance, the function \(f(x) = x^3 - 3x\) has three x-intercepts but only one point with a horizontal tangent.

Step 1: Understanding polynomial functions and their x-intercepts

We know that a polynomial function of degree \(n\) can have up to \(n\) x-intercepts, each corresponding to a real root. So, a function that has three x-intercepts is of at least degree three, but could be higher.

Step 2: Understanding horizontal tangents and local extremums

A tangent line to a function is horizontal at a point if the derivative of the function is equal to zero at that point. These points can also be described as local maximum or minimum points, depending on the area surrounding it.

Step 3: Verifying the given statement

Although a third degree polynomial can have three roots, it doesn't guarantee that it will have at least two points with horizontal tangents. For example, the cubic function \(f(x) = x^3 - 3x\) has three x-intercepts (at x = -1, 0, and 1), but there's only one point with a horizontal tangent (at x = 0). Therefore, the statement is false.

Step 4: Counterexample and Explanation

The counterexample provided contradicts the given statement. While it is possible for a function with three x-intercepts to have two points with a horizontal tangent, it's not mandatory. This can depend on the degree of the function, the type of roots, etc., and is therefore false without additional information.