Problem 249 - Calculus Volume 1

Question

For the following exercises, consider a third-degree polynomial \(f(x), \quad\) which has the properties \(f^{\prime}(1)=0, f^{\prime}(3)=0\) . Determine whether the following statements are true or false. Justify your answer. If \(f(x)\) has three roots, then it has 1 inflection point.

Step-by-Step Solution

Answer

True, because a third-degree polynomial always has exactly one inflection point.

Step 1: Understand the Given Information

We have a third-degree polynomial \( f(x) \). Two of its derivatives, \( f'(1) = 0 \) and \( f'(3) = 0 \), indicate that these are critical points where the slope of the tangent is zero. This suggests that these points are either local maxima, minima, or points of inflection.

Step 2: Analyze the Nature of Critical Points

Since a third-degree polynomial can have up to two critical points, and we are given that these are \( x = 1 \) and \( x = 3 \), these are the only critical points for the polynomial, assuming things like factors of multiplicity greater than 1 do not affect this.

Step 3: Consider the Properties of Third-Degree Polynomials

A third-degree polynomial can have up to three roots, corresponding to its degree. However, it will always have exactly one point of inflection since a cubic polynomial's second derivative, which would be quadratic, has one root.

Step 4: Connect Roots and Inflection Points

The presence of three roots in a cubic polynomial \( f(x) \) does not directly inform us about the number of inflection points. Regardless of the root configuration, a cubic polynomial will always have exactly one inflection point, which is determined by setting its second derivative equal to zero.

Step 5: Conclusion

Thus, whether \( f(x) \) has three roots or not, a third-degree polynomial will always have exactly one inflection point. The statement that if \( f(x) \) has three roots then it has 1 inflection point is true. However, the presence of three roots is irrelevant to the existence of one inflection point.

Key Concepts

Critical Points

Critical points are locations on a function where its derivative equals zero. This typically means the slope of the tangent line to the function at these points is flat. They signify potential places for local maxima, minima, or saddle points.

• Local maxima are points where the function reaches a peak and then decreases.
• Local minima are points where the function reaches a low and then increases.
• Saddle points are neither maxima nor minima but still have a zero derivative.

In the context of the given polynomial, the critical points are at \( x = 1 \) and \( x = 3 \), where the derivative \( f'(x) \) equals zero. These are the only points we need to consider for this exercise.

Derivative

A derivative represents the rate of change of a function with respect to its variable. In simpler terms, it tells us how steep the function is at any point.

• First derivative \( f'(x) \) provides the slope of the tangent line to the curve.
• Second derivative \( f''(x) \) tells us about the concavity of the function, showing where the function curves upwards or downwards.

For a third-degree polynomial, the first derivative is a quadratic function. The zeros of this derivative give us the critical points of the original polynomial, as seen with our steps using \( f'(1) = 0 \) and \( f'(3) = 0 \). Understanding derivatives helps in predicting the behavior of the function at various points.

Inflection Point

The inflection point of a function is where the curve changes concavity, shifting from curving upwards to curving downwards, or vice versa. This is found by solving \( f''(x) = 0 \), which in the case of a cubic function will give us one real solution.

• Inflection points highlight the flexibility in the graph's path.
• In a cubic polynomial, the second derivative is linear and thus, crosses the x-axis at a single point.

Regardless of the number of roots or critical points, a third-degree polynomial, such as that described in the original exercise, will always exhibit one inflection point.

Roots of Polynomial

Roots of a polynomial are the values of \( x \) that make the polynomial equal to zero. They represent the points where the function intersects the x-axis.

• A cubic polynomial can have up to three real roots.
• Roots are key in determining the shape and behavior of the polynomial graph.

In this exercise, having three roots does not affect the number of inflection points. While roots and inflection points are interrelated in defining a function's behavior, the presence of three roots ensures the polynomial intersects the x-axis three times, but it will still have one inflection point as dictated by its degree.