A function is a fifth-degree polynomial. How many turning points can it have?
A. exactly four
B. exactly five
C. four or less
D. five or less
Answer (1)
A turning point is where a function changes from increasing to decreasing or vice versa.
The number of turning points of a polynomial of degree n is at most n − 1 .
A fifth-degree polynomial has a derivative of degree 4, which can have at most 4 real roots.
Therefore, a fifth-degree polynomial can have four or less turning points: four or less .
Explanation
Understanding Turning Points A turning point of a function is a point where the function changes from increasing to decreasing (a local maximum) or from decreasing to increasing (a local minimum). The number of turning points of a polynomial function is related to its degree.
Turning Points and Derivatives The derivative of a polynomial of degree n is a polynomial of degree n − 1 . The turning points occur where the derivative is equal to zero. A polynomial of degree n − 1 can have at most n − 1 real roots. Therefore, a polynomial of degree n can have at most n − 1 turning points.
Applying to the Fifth-Degree Polynomial Since the given function is a fifth-degree polynomial (degree 5), its derivative will be a fourth-degree polynomial (degree 4). A fourth-degree polynomial can have at most 4 real roots. Therefore, the fifth-degree polynomial can have at most 4 turning points. It can also have fewer than 4 turning points.
Conclusion Therefore, a fifth-degree polynomial can have four or less turning points.
Examples
Consider a roller coaster design. The track can be modeled as a polynomial function, where the hills and valleys represent turning points. A fifth-degree polynomial allows for a track with up to four turning points, creating a thrilling ride with multiple changes in direction. Understanding the relationship between the degree of the polynomial and the number of turning points helps engineers design exciting and safe roller coasters.
