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Which of the following statements holds in (0, 2) if the function y = In (3x4 – 2x3 – 6x2 + 6x + 1)
- a)Maximum of y is In(39/16)
- b)f has neither maximum nor minimum (0, 2)
- c)f has maximum but not minimum in (0, 2)
- d)minimum value of y is in 2
Correct answer is option 'A,D'. Can you explain this answer?
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Verified Answer
Which of the following statements holds in (0, 2) if the function y = ...
Now for maximum or minimum of z, dz/dx = 0
Therefore z is maximum and minimum at x = 1/2 and x = 1 respectively
Hence y is also maximum and minimum at x = 1/2 and x = 1 respectively..
Maximum value of y is 
and minimum value of y is
In(3 – 2 – 6 + 6 + 1) i.e., In 2
and minimum value of y isIn(3 – 2 – 6 + 6 + 1) i.e., In 2
Most Upvoted Answer
Which of the following statements holds in (0, 2) if the function y = ...
Statement: The function y = In(3x^4 - 2x^3 + 6x^2 - 6x + 1) holds in the interval (0, 2).
Explanation:
To determine the behavior of the function in the given interval, we need to analyze its critical points, extrema, and concavity.
Finding the critical points:
Critical points occur when the derivative of the function is equal to zero or undefined. Let's first find the derivative of the given function:
f(x) = In(3x^4 - 2x^3 + 6x^2 - 6x + 1)
Using the chain rule and the fact that the derivative of ln(u) is (1/u) * u', we have:
f'(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * (12x^3 - 6x^2 + 12x - 6)
Now, setting f'(x) equal to zero and solving for x:
(12x^3 - 6x^2 + 12x - 6) = 0
Unfortunately, this equation does not have any simple solutions. We can use numerical methods or a graphing tool to find the approximate values of x where f'(x) = 0. Let's assume these values are x1, x2, and x3.
Identifying extrema:
To determine whether the critical points are extrema (minimum or maximum), we need to analyze the second derivative of the function.
f''(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * ((36x^2 - 12x + 12) * (3x^4 - 2x^3 + 6x^2 - 6x + 1) - (12x^3 - 6x^2 + 12x - 6) * (12x^3 - 6x^2 + 12x - 6))
Simplifying the expression, we have:
f''(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * (-36x^4 + 48x^3 - 24x^2 + 24)
Concavity:
To determine the concavity of the function, we need to analyze the sign of the second derivative in the interval (0, 2). We can do this by evaluating f''(x) at some points within the interval.
Evaluating f''(x) at x = 1:
f''(1) = (1 / (3(1)^4 - 2(1)^3 + 6(1)^2 - 6(1) + 1)) * (-36(1)^4 + 48(1)^3 - 24(1)^2 + 24)
= (1 / (3 - 2 + 6 - 6 + 1)) * (-36 + 48 - 24 + 24)
= (1 /
Explanation:
To determine the behavior of the function in the given interval, we need to analyze its critical points, extrema, and concavity.
Finding the critical points:
Critical points occur when the derivative of the function is equal to zero or undefined. Let's first find the derivative of the given function:
f(x) = In(3x^4 - 2x^3 + 6x^2 - 6x + 1)
Using the chain rule and the fact that the derivative of ln(u) is (1/u) * u', we have:
f'(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * (12x^3 - 6x^2 + 12x - 6)
Now, setting f'(x) equal to zero and solving for x:
(12x^3 - 6x^2 + 12x - 6) = 0
Unfortunately, this equation does not have any simple solutions. We can use numerical methods or a graphing tool to find the approximate values of x where f'(x) = 0. Let's assume these values are x1, x2, and x3.
Identifying extrema:
To determine whether the critical points are extrema (minimum or maximum), we need to analyze the second derivative of the function.
f''(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * ((36x^2 - 12x + 12) * (3x^4 - 2x^3 + 6x^2 - 6x + 1) - (12x^3 - 6x^2 + 12x - 6) * (12x^3 - 6x^2 + 12x - 6))
Simplifying the expression, we have:
f''(x) = (1 / (3x^4 - 2x^3 + 6x^2 - 6x + 1)) * (-36x^4 + 48x^3 - 24x^2 + 24)
Concavity:
To determine the concavity of the function, we need to analyze the sign of the second derivative in the interval (0, 2). We can do this by evaluating f''(x) at some points within the interval.
Evaluating f''(x) at x = 1:
f''(1) = (1 / (3(1)^4 - 2(1)^3 + 6(1)^2 - 6(1) + 1)) * (-36(1)^4 + 48(1)^3 - 24(1)^2 + 24)
= (1 / (3 - 2 + 6 - 6 + 1)) * (-36 + 48 - 24 + 24)
= (1 /
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Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer?
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Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer?.
Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer? for IIT JAM 2024 is part of IIT JAM preparation. The Question and answers have been prepared according to the IIT JAM exam syllabus. Information about Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer? covers all topics & solutions for IIT JAM 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Which of the following statements holds in (0, 2) if the function y = In (3x4 2x3 6x2 + 6x + 1)a)Maximum of y is In(39/16)b)f has neither maximum nor minimum (0, 2)c)f has maximum but not minimum in (0, 2)d)minimum value of y is in 2Correct answer is option 'A,D'. Can you explain this answer?.
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