Question

Polynomial p(x)=x3−ax: X-intercepts: To find the x-intercepts, set p(x) equal to zero and solve for x: x3−ax=0 Factoring out x, we get x(x2−a)=0. So, there is one x-intercept at x=0 and two additional x-intercepts at x=a​ and x=−a​ if a>0 (since a>0, a​ is real). Local extrema: To find the local extrema, take the derivative of p(x) with respect to x and set it equal to zero: p′(x)=3x2−a Setting 3x2−a=0 gives x=3a​​ and x=−3a​​. We need to check the sign of p′(x) around these points to determine the nature of the extrema. If a>0, then p′(x)>0 for x<−3a​​, p′(x)<0 for −3a​​0 for x>3a​​. Therefore, there is a local minimum at x=−3a​​ and a local maximum at x=3a​​. Polynomial q(x)=x3−ax: The analysis for q(x) is the same as for p(x) since they are the same polynomial. In summary: Both p(x) and q(x) have three x-intercepts when a>0, located at x=0, x=a​, and x=−a​. Both p(x) and q(x) have one local minimum at x=−3a​​ and one local maximum at x=3a​​. The correct multiple-choice answer could be: a) 3 x-intercepts, 1 local minimum, and 1 local maximum. b) 3 x-intercepts, 2 local minima, and 2 local maxima. c) 3 x-intercepts, 1 local minimum, and 2 local maxima. d) 3 x-intercepts, 2 local minima, and 1 local maximum.

Answer 1

For the given polynomial p(x)=x3−ax with a>0: It has three x-intercepts at x=0,x=a, and x=−a. There is one local minimum at x=−3a and two local maxima at x=3a. This leads to the correct answer: c) 3 x-intercepts, 1 local minimum, and 2 local maxima.

This correct answer is c)

Step-by-step explanation:

Step-by-step explanation:

1. X-intercepts:

The polynomial p(x)=x3−ax has three x-intercepts when a>0: x=0, x=a, and x=−a.

2. Local Extrema:

• The derivative of p(x) is p′ (x)=3x2−a.

• Setting 3x2−a=0 gives x= √a/3 and x= -√a/3.

• Checking the sign of p′ (x) around these points, we find:

• If a>0, then p′ (x)>0 for x<− √a/3, p′ (x)<0 for − √a/3 <x< √a/3, and p′ (x)>0 for x> √a/3

Therefore, there is a local minimum at x=− √a/3 and a local maximum at

x= √a/3.

In summary, the characteristics of the polynomial p(x)=x3−ax match the description in option c).

This correct answer is c)