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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.
Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.
Reason (R): A strictly increasing function is an injective function.
- a)Both A and R are true and R is the correct explanation of A
- b)Both A and R are true but R is NOT the correct explanation of A
- c)A is true but R is false
- d)A is false and R is True
Correct answer is option 'B'. Can you explain this answer?
Verified Answer
Directions : In the following questions, A statement of Assertion (A)...
f(x) = x3 - 3x2+ 6x - 100
f(x) = 3x2 - 6x + 6
= 3[x2 - 2x + 2]
= 3[(x-1)2+1]
Since f’(x) > 0; x ∊ R
f(x) is strictly increasing on R.
Hence A is true.
For a strictly increasing function,
x1 > x2⇒ f(x1) > f(x2)
i.e.) x1 = x2
⇒ f(x1) — f(x2)
Hence, a strictly increasing function is always an injective function.
So R is true.
But R is not the correct explanation of A.
Most Upvoted Answer
Directions : In the following questions, A statement of Assertion (A)...
Assertion (A): The function f(x) = x^3 – 3x^2 + 6x – 100 is strictly increasing on the set of real numbers.
Reason (R): A strictly increasing function is an injective function.
To determine whether Assertion (A) and Reason (R) are true or false, we need to analyze each statement separately.
Analysis of Assertion (A):
To check whether the function f(x) = x^3 – 3x^2 + 6x – 100 is strictly increasing, we need to examine the sign of its derivative. If the derivative is positive for all values of x, then the function is strictly increasing.
Taking the derivative of f(x), we get:
f'(x) = 3x^2 – 6x + 6
To find the critical points of the function, we set f'(x) = 0:
3x^2 – 6x + 6 = 0
Using the quadratic formula, we get:
x = (6 ± √(6^2 - 4(3)(6))) / (2(3))
x = (6 ± √(36 - 72)) / 6
x = (6 ± √(-36)) / 6
x = (6 ± 6i√(1)) / 6
x = 1 ± i
Since the derivative does not equal zero for any real value of x, there are no critical points. Therefore, the derivative is always positive, indicating that the function is strictly increasing on the set of real numbers.
Analysis of Reason (R):
The reason states that a strictly increasing function is an injective function. This is true because if a function is strictly increasing, it means that for any two distinct input values x1 and x2, the corresponding output values f(x1) and f(x2) will also be distinct. In other words, the function does not map different inputs to the same output, making it injective.
Conclusion:
Both Assertion (A) and Reason (R) are true. Furthermore, Reason (R) correctly explains Assertion (A) because a strictly increasing function is indeed an injective function. Therefore, option B is the correct answer.
Reason (R): A strictly increasing function is an injective function.
To determine whether Assertion (A) and Reason (R) are true or false, we need to analyze each statement separately.
Analysis of Assertion (A):
To check whether the function f(x) = x^3 – 3x^2 + 6x – 100 is strictly increasing, we need to examine the sign of its derivative. If the derivative is positive for all values of x, then the function is strictly increasing.
Taking the derivative of f(x), we get:
f'(x) = 3x^2 – 6x + 6
To find the critical points of the function, we set f'(x) = 0:
3x^2 – 6x + 6 = 0
Using the quadratic formula, we get:
x = (6 ± √(6^2 - 4(3)(6))) / (2(3))
x = (6 ± √(36 - 72)) / 6
x = (6 ± √(-36)) / 6
x = (6 ± 6i√(1)) / 6
x = 1 ± i
Since the derivative does not equal zero for any real value of x, there are no critical points. Therefore, the derivative is always positive, indicating that the function is strictly increasing on the set of real numbers.
Analysis of Reason (R):
The reason states that a strictly increasing function is an injective function. This is true because if a function is strictly increasing, it means that for any two distinct input values x1 and x2, the corresponding output values f(x1) and f(x2) will also be distinct. In other words, the function does not map different inputs to the same output, making it injective.
Conclusion:
Both Assertion (A) and Reason (R) are true. Furthermore, Reason (R) correctly explains Assertion (A) because a strictly increasing function is indeed an injective function. Therefore, option B is the correct answer.
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?
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Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? for JEE 2025 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer?.
Solutions for Directions : In the following questions, A statement of Assertion (A) is followed by a statement of Reason (R). Mark the correct choice as.Assertion (A): The function f(x) = x3 – 3x2 + 6x – 100 is strictly increasing on the set of real numbers.Reason (R): A strictly increasing function is an injective function.a)Both A and R are true and R is the correct explanation of Ab)Both A and R are true but R is NOT the correct explanation of Ac)A is true but R is falsed)A is false and R is TrueCorrect answer is option 'B'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
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