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Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ?
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Let p(x) be a real polynomial of least degree which has local maximum ...
Problem statement:
Let p(x) be a real polynomial of least degree which has a local maximum at x=1 and a local minimum at x=3. If p(1)=6 and p(3)=2, find p'(0).
Solution:
Step 1: Identify the key points given in the problem:
- The polynomial p(x) has a local maximum at x=1.
- The polynomial p(x) has a local minimum at x=3.
- p(1)=6 and p(3)=2.
Step 2: Understanding local maximum and minimum:
- A local maximum occurs at a point where the function reaches the highest value in a specific region.
- A local minimum occurs at a point where the function reaches the lowest value in a specific region.
Step 3: Understanding the behavior of the polynomial:
- Since p(x) has a local maximum at x=1 and a local minimum at x=3, the behavior of the polynomial changes around these points.
- As x approaches 1 from the left side, the polynomial increases.
- As x approaches 1 from the right side, the polynomial decreases.
- As x approaches 3 from the left side, the polynomial decreases.
- As x approaches 3 from the right side, the polynomial increases.
Step 4: Sketching the polynomial:
- Based on the given information, we can sketch a rough graph of the polynomial.
- The graph will have a local maximum at x=1 and a local minimum at x=3.
- The y-value at x=1 is 6 and the y-value at x=3 is 2.
Step 5: Determining the degree of the polynomial:
- The problem states that p(x) is a real polynomial of least degree.
- Since we have two key points (x=1 and x=3) and two corresponding y-values (6 and 2), the polynomial is at least a quadratic polynomial.
Step 6: Finding p'(0):
- To find p'(0), we need to find the derivative of the polynomial and evaluate it at x=0.
- Since p(x) is at least a quadratic polynomial, let's assume it is a quadratic polynomial of the form p(x) = ax^2 + bx + c.
- Taking the derivative of p(x) with respect to x, we get p'(x) = 2ax + b.
- Evaluating p'(x) at x=0, we get p'(0) = 2a(0) + b = b.
Step 7: Conclusion:
- From step 6, we found that p'(0) = b.
- However, we do not have enough information to determine the value of b or p'(0) based on the given information.
- Therefore, we cannot determine the value of p'(0) with the given information.
Let p(x) be a real polynomial of least degree which has a local maximum at x=1 and a local minimum at x=3. If p(1)=6 and p(3)=2, find p'(0).
Solution:
Step 1: Identify the key points given in the problem:
- The polynomial p(x) has a local maximum at x=1.
- The polynomial p(x) has a local minimum at x=3.
- p(1)=6 and p(3)=2.
Step 2: Understanding local maximum and minimum:
- A local maximum occurs at a point where the function reaches the highest value in a specific region.
- A local minimum occurs at a point where the function reaches the lowest value in a specific region.
Step 3: Understanding the behavior of the polynomial:
- Since p(x) has a local maximum at x=1 and a local minimum at x=3, the behavior of the polynomial changes around these points.
- As x approaches 1 from the left side, the polynomial increases.
- As x approaches 1 from the right side, the polynomial decreases.
- As x approaches 3 from the left side, the polynomial decreases.
- As x approaches 3 from the right side, the polynomial increases.
Step 4: Sketching the polynomial:
- Based on the given information, we can sketch a rough graph of the polynomial.
- The graph will have a local maximum at x=1 and a local minimum at x=3.
- The y-value at x=1 is 6 and the y-value at x=3 is 2.
Step 5: Determining the degree of the polynomial:
- The problem states that p(x) is a real polynomial of least degree.
- Since we have two key points (x=1 and x=3) and two corresponding y-values (6 and 2), the polynomial is at least a quadratic polynomial.
Step 6: Finding p'(0):
- To find p'(0), we need to find the derivative of the polynomial and evaluate it at x=0.
- Since p(x) is at least a quadratic polynomial, let's assume it is a quadratic polynomial of the form p(x) = ax^2 + bx + c.
- Taking the derivative of p(x) with respect to x, we get p'(x) = 2ax + b.
- Evaluating p'(x) at x=0, we get p'(0) = 2a(0) + b = b.
Step 7: Conclusion:
- From step 6, we found that p'(0) = b.
- However, we do not have enough information to determine the value of b or p'(0) based on the given information.
- Therefore, we cannot determine the value of p'(0) with the given information.
Community Answer
Let p(x) be a real polynomial of least degree which has local maximum ...
Introduction:
We are given a real polynomial p(x) of least degree that has a local maximum at x=1 and a local minimum at x=3. We are also given that p(1)=6 and p(3)=2. We need to find the value of p'(0), which represents the derivative of p(x) at x=0.
Approach:
To find the value of p'(0), we need to analyze the behavior of the polynomial p(x) around x=0. Since p(x) has a local maximum at x=1 and a local minimum at x=3, we can infer some information about the shape of the polynomial.
Key Points:
1. A local maximum or minimum occurs when the derivative of a function changes sign from positive to negative (for a maximum) or negative to positive (for a minimum).
2. The degree of the polynomial p(x) is the highest power of x in the polynomial.
3. The derivative of a polynomial of degree n is a polynomial of degree n-1.
Analysis:
1. Since p(x) has a local maximum at x=1, the derivative of p(x) changes sign from positive to negative at x=1.
2. Similarly, since p(x) has a local minimum at x=3, the derivative of p(x) changes sign from negative to positive at x=3.
3. Since p(x) is a real polynomial of least degree, it must be a quadratic polynomial (degree 2) because a linear polynomial (degree 1) cannot have both a local maximum and minimum.
4. Therefore, the derivative of p(x) is a linear polynomial (degree 1).
5. Since the derivative of p(x) changes sign from positive to negative at x=1, and from negative to positive at x=3, it means the derivative has a root (zero) between x=1 and x=3.
6. Therefore, the derivative of p(x) can be written as p'(x) = ax + b, where a and b are constants.
7. We are given that p(1)=6, which means when x=1, p(x) equals 6. This gives us the equation p(1) = a + b = 6.
8. We are also given that p(3)=2, which means when x=3, p(x) equals 2. This gives us the equation p(3) = 3a + b = 2.
9. Solving these two equations simultaneously, we can find the values of a and b.
10. Once we have the values of a and b, we can evaluate p'(0) by substituting x=0 in the expression p'(x) = ax + b.
Conclusion:
By analyzing the behavior of the polynomial p(x) and using the given conditions, we can find the values of a and b, and hence evaluate p'(0).
We are given a real polynomial p(x) of least degree that has a local maximum at x=1 and a local minimum at x=3. We are also given that p(1)=6 and p(3)=2. We need to find the value of p'(0), which represents the derivative of p(x) at x=0.
Approach:
To find the value of p'(0), we need to analyze the behavior of the polynomial p(x) around x=0. Since p(x) has a local maximum at x=1 and a local minimum at x=3, we can infer some information about the shape of the polynomial.
Key Points:
1. A local maximum or minimum occurs when the derivative of a function changes sign from positive to negative (for a maximum) or negative to positive (for a minimum).
2. The degree of the polynomial p(x) is the highest power of x in the polynomial.
3. The derivative of a polynomial of degree n is a polynomial of degree n-1.
Analysis:
1. Since p(x) has a local maximum at x=1, the derivative of p(x) changes sign from positive to negative at x=1.
2. Similarly, since p(x) has a local minimum at x=3, the derivative of p(x) changes sign from negative to positive at x=3.
3. Since p(x) is a real polynomial of least degree, it must be a quadratic polynomial (degree 2) because a linear polynomial (degree 1) cannot have both a local maximum and minimum.
4. Therefore, the derivative of p(x) is a linear polynomial (degree 1).
5. Since the derivative of p(x) changes sign from positive to negative at x=1, and from negative to positive at x=3, it means the derivative has a root (zero) between x=1 and x=3.
6. Therefore, the derivative of p(x) can be written as p'(x) = ax + b, where a and b are constants.
7. We are given that p(1)=6, which means when x=1, p(x) equals 6. This gives us the equation p(1) = a + b = 6.
8. We are also given that p(3)=2, which means when x=3, p(x) equals 2. This gives us the equation p(3) = 3a + b = 2.
9. Solving these two equations simultaneously, we can find the values of a and b.
10. Once we have the values of a and b, we can evaluate p'(0) by substituting x=0 in the expression p'(x) = ax + b.
Conclusion:
By analyzing the behavior of the polynomial p(x) and using the given conditions, we can find the values of a and b, and hence evaluate p'(0).
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Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ?
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Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ?.
Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ? for Mathematics 2024 is part of Mathematics preparation. The Question and answers have been prepared according to the Mathematics exam syllabus. Information about Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ? covers all topics & solutions for Mathematics 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let p(x) be a real polynomial of least degree which has local maximum at x=1 and local min at x= 3.if p(1)=6 and p(3)=2 then p'(0) is ?.
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