Question
Rogelio is asked to sketch a graph of g(x)=3x^3(x−5)^2(5−x)^4 . How many turning points should his sketch include?
Answers
GPT-4o mini
To find out how many turning points the function \( g(x) = 3x^3(x-5)^2(5-x)^4 \) has, we need to determine the derivative of the function and analyze its critical points.
1. **Find the degree of the polynomial**: The expression consists of several factors:
- \( 3x^3 \) contributes a degree of 3.
- \( (x - 5)^2 \) contributes a degree of 2.
- \( (5 - x)^4 \) contributes a degree of 4.
Combining these, the total degree of \( g(x) \) is:
\[
3 + 2 + 4 = 9
\]
2. **The maximum number of turning points**: The maximum number of turning points in a polynomial function is \( n-1 \), where \( n \) is the degree of the polynomial. For our polynomial, which is of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]
3. **Multiplicity of roots and their effects**: To accurately count the turning points, we need to consider the roots of \( g(x) \) and their multiplicities:
- The function has roots at \( x = 0 \) (multiplicity 3), \( x = 5 \) (multiplicity 2), and \( x = 5 \) (from \( (5-x)^4 \), which is the same point, contributing a total of \( 4 \) more).
- Thus, the roots of \( g(x) \) and their total multiplicities are:
- \( x = 0 \): multiplicity 3 (thus contributes only 2 turning points, as turning points occur only at odd multiplicities).
- \( x = 5 \): multiplicity 2 (does not contribute a turning point).
- \( (5-x)^4 \) at \( x=5 \): contributes no additional turning points.
4. **Analyzing contributions**: The effective multiplicities show that:
- At \( x=0 \), with a multiplicity of 3, there will be 2 turning points (because an odd multiplicity results in a turning point, but the even parts indicate no additional crossing).
- At \( x=5 \), both multiplicities (2 from \( (x-5)^2 \) and 4 from \( (5-x)^4 \)), being even, do not contribute any additional turning points.
5. **Summary**: Since the only point contributing turning points is \( x=0 \) (2 turning points), we conclude:
- Total turning points = 2.
Thus, the sketch of \( g(x) \) should include a total of **2 turning points**.
1. **Find the degree of the polynomial**: The expression consists of several factors:
- \( 3x^3 \) contributes a degree of 3.
- \( (x - 5)^2 \) contributes a degree of 2.
- \( (5 - x)^4 \) contributes a degree of 4.
Combining these, the total degree of \( g(x) \) is:
\[
3 + 2 + 4 = 9
\]
2. **The maximum number of turning points**: The maximum number of turning points in a polynomial function is \( n-1 \), where \( n \) is the degree of the polynomial. For our polynomial, which is of degree 9, the maximum number of turning points is:
\[
9 - 1 = 8
\]
3. **Multiplicity of roots and their effects**: To accurately count the turning points, we need to consider the roots of \( g(x) \) and their multiplicities:
- The function has roots at \( x = 0 \) (multiplicity 3), \( x = 5 \) (multiplicity 2), and \( x = 5 \) (from \( (5-x)^4 \), which is the same point, contributing a total of \( 4 \) more).
- Thus, the roots of \( g(x) \) and their total multiplicities are:
- \( x = 0 \): multiplicity 3 (thus contributes only 2 turning points, as turning points occur only at odd multiplicities).
- \( x = 5 \): multiplicity 2 (does not contribute a turning point).
- \( (5-x)^4 \) at \( x=5 \): contributes no additional turning points.
4. **Analyzing contributions**: The effective multiplicities show that:
- At \( x=0 \), with a multiplicity of 3, there will be 2 turning points (because an odd multiplicity results in a turning point, but the even parts indicate no additional crossing).
- At \( x=5 \), both multiplicities (2 from \( (x-5)^2 \) and 4 from \( (5-x)^4 \)), being even, do not contribute any additional turning points.
5. **Summary**: Since the only point contributing turning points is \( x=0 \) (2 turning points), we conclude:
- Total turning points = 2.
Thus, the sketch of \( g(x) \) should include a total of **2 turning points**.