Question

Consider the polynomial p(x)=x ^5 −3x ^4 +3x ^3−x^ 2. Match each value with the label that best describes it.

A. Local maximum
B. Local minimum
C. Point of inflection
D. Constant term
E. Degree of the polynomial

Answer 1

The degree of the polynomial p(x) is 5 as it's the highest exponent, and the constant term is zero since it's missing in the equation. Calculating the local maximum, local minimum, and points of inflection requires derivative analysis which isn't provided here.

Step-by-step explanation:

To match each value with the appropriate label for the polynomial p(x)=x^5−3x^4+3x^3−x^2:

• E. Degree of the polynomial is 5, as that is the highest power of x in the polynomial.
• D. Constant term in this polynomial is absent, which implies that the constant term is zero.
• Identifying a A. Local maximum or B. Local minimum would require finding the first and second derivatives of p(x) to assess the critical points and their nature based on the sign of the second derivative.
• A C. Point of inflection is where the concavity of the graph changes, which also can be determined from the signs of the second derivative around a point.

Without actually calculating the derivatives, which would be necessary to find exact local maxima, minima, or points of inflection, we are unable to accurately match these labels solely based on the given polynomial. However, we can confirm the degree and the absence of a constant term.