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Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0?
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Which of the following statements are true when phi(x) is a polynomial...
True Statements for Polynomial phi(x)
1. At least one root of phi'(x) between any two roots of phi(x)=0
- This statement is true for all polynomial functions since the derivative of a polynomial function will always have one less degree than the original function. Therefore, if there are n roots of phi(x)=0, then there will be at least n-1 roots of phi'(x)=0 between them.
2. At least one root of x phi'(x) between any two roots of phi(x)=0, phi(x)=0
- This statement is also true for all polynomial functions. We can rewrite x phi'(x) as the derivative of x^2/2 phi(x), and by the Mean Value Theorem, we know that there exists a point c between any two roots of phi(x)=0 such that x^2/2 phi'(c) = (b-a) phi(b) for some a and b in the interval. Therefore, we have x phi'(c) = 2(b-a) phi(b)/x, which implies that there exists at least one root of x phi'(x) between any two roots of phi(x)=0.
3. At least one root of (x^2+1) phi'(x) between any two roots of phi(x)=0, x=0
- This statement is not necessarily true for all polynomial functions. However, it is true for all odd degree polynomial functions since they have at least one real root. We can use a similar argument as in statement 2 to show that there exists a root of (x^2+1) phi'(x) between any two roots of phi(x)=0, x=0.
4. At least one root of phi'(x) xlamda(x) between any two roots of phi(x)=0
- This statement is equivalent to statement 1 and is therefore true for all polynomial functions.
In summary, statements 1 and 4 are always true for polynomial functions, while statements 2 and 3 are true for all polynomial functions except for those of even degree.
1. At least one root of phi'(x) between any two roots of phi(x)=0
- This statement is true for all polynomial functions since the derivative of a polynomial function will always have one less degree than the original function. Therefore, if there are n roots of phi(x)=0, then there will be at least n-1 roots of phi'(x)=0 between them.
2. At least one root of x phi'(x) between any two roots of phi(x)=0, phi(x)=0
- This statement is also true for all polynomial functions. We can rewrite x phi'(x) as the derivative of x^2/2 phi(x), and by the Mean Value Theorem, we know that there exists a point c between any two roots of phi(x)=0 such that x^2/2 phi'(c) = (b-a) phi(b) for some a and b in the interval. Therefore, we have x phi'(c) = 2(b-a) phi(b)/x, which implies that there exists at least one root of x phi'(x) between any two roots of phi(x)=0.
3. At least one root of (x^2+1) phi'(x) between any two roots of phi(x)=0, x=0
- This statement is not necessarily true for all polynomial functions. However, it is true for all odd degree polynomial functions since they have at least one real root. We can use a similar argument as in statement 2 to show that there exists a root of (x^2+1) phi'(x) between any two roots of phi(x)=0, x=0.
4. At least one root of phi'(x) xlamda(x) between any two roots of phi(x)=0
- This statement is equivalent to statement 1 and is therefore true for all polynomial functions.
In summary, statements 1 and 4 are always true for polynomial functions, while statements 2 and 3 are true for all polynomial functions except for those of even degree.
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Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0?
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Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0?.
Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0?.
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Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0?, a detailed solution for Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? has been provided alongside types of Which of the following statements are true when phi(x) is a polynomial? 1. Between any two roots of phi(x)=0, atleast one root of phi'(x) lamda. Phi(x) =0 2.Between any two roots of phi(x)=0, atleast one root of x phi'(x) lamda phi(x)=0 3. Between any two roots of phi(x)=0, atleast one root of (x^2 1) phi'(x) lamda x =0 Between any two roots of phi(x)=0, atleast one root of phi'(x) xlamda(x) =0? theory, EduRev gives you an
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