Answer :
To estimate the local minimum of the polynomial [tex]\( y = -5x^6 - 6x^5 + 50x^3 + 45x^2 - 108x - 108 \)[/tex], we need to analyze the behavior of the function at some potential critical points. Let's go through the process to identify the local minima.
### Steps to Find Local Minima:
1. Identify Potential Critical Points:
The problem provides specific critical points in the answer choices, which are [tex]\( x = -3 \)[/tex], [tex]\( x = 0.618 \)[/tex], and [tex]\( x = 0 \)[/tex].
2. Evaluate the Function at These Points:
Calculate the polynomial value at these critical points:
- [tex]\( y(-3) \)[/tex]
- [tex]\( y(0.618) \)[/tex]
- [tex]\( y(0) \)[/tex]
3. Apply the Second-Derivative Test:
To determine if these points are minima, use the second derivative test. Calculate the second derivative of the polynomial and evaluate it at the critical points. The second derivative will tell us the nature of the points:
- If the second derivative is positive at a point, the function has a local minimum there.
- If it's negative, the point is a local maximum.
- If it's zero, the test is inconclusive.
4. Determine the Local Minimum:
Among the calculations:
- At [tex]\( x = 0.618 \)[/tex], the calculated value of the polynomial is approximately [tex]\( y \approx -146.353 \)[/tex], and the second derivative indicates a minimum.
- At [tex]\( x = 0 \)[/tex], the polynomial value is [tex]\( y = -108 \)[/tex], and the nature of the point is also a minimum.
Based on the evaluations and considering the given options:
- Point [tex]\( B \)[/tex] [tex]\((0.618, -146.353)\)[/tex] corresponds to a local minimum due to both the value calculation and behavior of the function at this point.
Therefore, the local minimum of the polynomial is choice B. [tex]\((0.618, -146.353)\)[/tex].
### Steps to Find Local Minima:
1. Identify Potential Critical Points:
The problem provides specific critical points in the answer choices, which are [tex]\( x = -3 \)[/tex], [tex]\( x = 0.618 \)[/tex], and [tex]\( x = 0 \)[/tex].
2. Evaluate the Function at These Points:
Calculate the polynomial value at these critical points:
- [tex]\( y(-3) \)[/tex]
- [tex]\( y(0.618) \)[/tex]
- [tex]\( y(0) \)[/tex]
3. Apply the Second-Derivative Test:
To determine if these points are minima, use the second derivative test. Calculate the second derivative of the polynomial and evaluate it at the critical points. The second derivative will tell us the nature of the points:
- If the second derivative is positive at a point, the function has a local minimum there.
- If it's negative, the point is a local maximum.
- If it's zero, the test is inconclusive.
4. Determine the Local Minimum:
Among the calculations:
- At [tex]\( x = 0.618 \)[/tex], the calculated value of the polynomial is approximately [tex]\( y \approx -146.353 \)[/tex], and the second derivative indicates a minimum.
- At [tex]\( x = 0 \)[/tex], the polynomial value is [tex]\( y = -108 \)[/tex], and the nature of the point is also a minimum.
Based on the evaluations and considering the given options:
- Point [tex]\( B \)[/tex] [tex]\((0.618, -146.353)\)[/tex] corresponds to a local minimum due to both the value calculation and behavior of the function at this point.
Therefore, the local minimum of the polynomial is choice B. [tex]\((0.618, -146.353)\)[/tex].
