JEE Exam > JEE Questions > The function f(x) = 2x3 – 3x2 – 1...
Start Learning for Free
The function f(x) = 2x3 – 3x2 – 12x + 4, has
- a)two points of local maximum
- b)two points of local minimum
- c)one maxima and one minima
- d)no maxima or minima
Correct answer is option 'C'. Can you explain this answer?
| FREE This question is part of | Download PDF Attempt this Test |
Most Upvoted Answer
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points o...
f(x) = 2x3 – 3x2 – 12x + 4
⇒ f '(x) = 6x2 – 6x – 12 = 6(x2 – x – 2)
= 6(x – 2) (x + 1)
For maxima and minima f '(x) = 0
∴ 6(x – 2)(x + 1) = 0 Þ x = 2, – 1
Now, f ''(x) = 12x-6
At x = 2; f ''(x) = 24 - 6 = 18> 0
∴ x = 2 , local min. point
At x = – 1; f ''(x) = 12 ( -1) - 6 = -18<0
∴ x = –1 local max. point
⇒ f '(x) = 6x2 – 6x – 12 = 6(x2 – x – 2)
= 6(x – 2) (x + 1)
For maxima and minima f '(x) = 0
∴ 6(x – 2)(x + 1) = 0 Þ x = 2, – 1
Now, f ''(x) = 12x-6
At x = 2; f ''(x) = 24 - 6 = 18> 0
∴ x = 2 , local min. point
At x = – 1; f ''(x) = 12 ( -1) - 6 = -18<0
∴ x = –1 local max. point
Free Test
FREE
| Start Free Test |
Community Answer
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points o...
Understanding the Function
The function given is f(x) = 2x^3 – 3x^2 – 12x + 4. To analyze its local maxima and minima, we first need to find the critical points by calculating the first derivative.
Finding the First Derivative
- The first derivative f'(x) = 6x^2 - 6x - 12.
- Setting f'(x) = 0 gives us the critical points.
Finding Critical Points
- We solve the equation: 6x^2 - 6x - 12 = 0.
- Simplifying, we get: x^2 - x - 2 = 0.
- Factoring provides the solutions: (x - 2)(x + 1) = 0, yielding x = 2 and x = -1.
Determining Local Extrema
- To classify the critical points, we use the second derivative test.
- The second derivative f''(x) = 12x - 6.
Evaluating the Second Derivative
- For x = 2: f''(2) = 12(2) - 6 = 24, which is positive. This indicates a local minimum.
- For x = -1: f''(-1) = 12(-1) - 6 = -18, which is negative. This indicates a local maximum.
Conclusion
- The function has one local maximum at x = -1 and one local minimum at x = 2.
- Hence, the correct answer is option 'C': one maxima and one minima.
This analysis shows that f(x) exhibits one peak and one trough, confirming the classification of its critical points effectively.
The function given is f(x) = 2x^3 – 3x^2 – 12x + 4. To analyze its local maxima and minima, we first need to find the critical points by calculating the first derivative.
Finding the First Derivative
- The first derivative f'(x) = 6x^2 - 6x - 12.
- Setting f'(x) = 0 gives us the critical points.
Finding Critical Points
- We solve the equation: 6x^2 - 6x - 12 = 0.
- Simplifying, we get: x^2 - x - 2 = 0.
- Factoring provides the solutions: (x - 2)(x + 1) = 0, yielding x = 2 and x = -1.
Determining Local Extrema
- To classify the critical points, we use the second derivative test.
- The second derivative f''(x) = 12x - 6.
Evaluating the Second Derivative
- For x = 2: f''(2) = 12(2) - 6 = 24, which is positive. This indicates a local minimum.
- For x = -1: f''(-1) = 12(-1) - 6 = -18, which is negative. This indicates a local maximum.
Conclusion
- The function has one local maximum at x = -1 and one local minimum at x = 2.
- Hence, the correct answer is option 'C': one maxima and one minima.
This analysis shows that f(x) exhibits one peak and one trough, confirming the classification of its critical points effectively.
Attention JEE Students!
To make sure you are not studying endlessly, EduRev has designed JEE study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in JEE.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer?
Question Description
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer?.
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? for JEE 2024 is part of JEE preparation. The Question and answers have been prepared according to the JEE exam syllabus. Information about The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer?.
Solutions for The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for JEE.
Download more important topics, notes, lectures and mock test series for JEE Exam by signing up for free.
Here you can find the meaning of The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice The function f(x) = 2x3 – 3x2 – 12x + 4, hasa)two points of local maximumb)two points of local minimumc)one maxima and one minimad)no maxima or minimaCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Suggested Free Tests
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
Follow Us
|
Signup on EduRev and stay on top of your study goals
10M+ students crushing their study goals daily