JEE Exam > JEE Questions > Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1...
Start Learning for Free
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1
- a)f(x) has a maximum
- b)f(x) has a minimum
- c)f'(x) has a maximum
- d)f'(x) has a minimum
Correct answer is option 'D'. Can you explain this answer?
Verified Answer
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x...
f(x) = (x+1)3 + 1 ∴ f'(x) = 3(x+1)2.
f'(x) = 0 ⇒ x = -1.
Now, f" (-1 - ∈) = 3(-∈)2 > 0, f'(-1 + ∈)2 = 3∈2 > 0.
∴ f(x) has neither a maximum nor a minimum at x = -1.
Let f'(x) = φ ′ (x) = 3(x+1)2 ∴ φ ′ (x) = 6(x+1).
φ ′ (x) = 0 ⇒ x = -1
φ ′ (-1-∈) = 6(-∈) < 0, φ ′ (-1-∈) = 6∈ > 0
∴ φ (x) has a minimum at x = -1
f'(x) = 0 ⇒ x = -1.
Now, f" (-1 - ∈) = 3(-∈)2 > 0, f'(-1 + ∈)2 = 3∈2 > 0.
∴ f(x) has neither a maximum nor a minimum at x = -1.
Let f'(x) = φ ′ (x) = 3(x+1)2 ∴ φ ′ (x) = 6(x+1).
φ ′ (x) = 0 ⇒ x = -1
φ ′ (-1-∈) = 6(-∈) < 0, φ ′ (-1-∈) = 6∈ > 0
∴ φ (x) has a minimum at x = -1
Most Upvoted Answer
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x...
Y' = 3x^2 + 6x + 3
if y' = 0
x^2 + 2x + 1 = 0
or,
(x + 1)^2 = 0
so,
x = -1
now y" = 6x + 6
at x = -1 , y" = 0
so f(x) has neither maximum and nor minimum.
Now f'(x) = g(x)consider as the main function then :-
g(x) = 3x^2 + 6x + 3
or,
g'(x) = 6x + 6
if g'(x) = 0
then 6x + 6 = 0
or, x = -1
now , g"(x) = 6
at x = -1 , g"(x) > 0 so f'(x) = g(x) has minimum value at x = -1 which is g(-1) = 3 - 6 + 3 = 0
Therefore we can easily say that the option (d) is the correct answer
if y' = 0
x^2 + 2x + 1 = 0
or,
(x + 1)^2 = 0
so,
x = -1
now y" = 6x + 6
at x = -1 , y" = 0
so f(x) has neither maximum and nor minimum.
Now f'(x) = g(x)consider as the main function then :-
g(x) = 3x^2 + 6x + 3
or,
g'(x) = 6x + 6
if g'(x) = 0
then 6x + 6 = 0
or, x = -1
now , g"(x) = 6
at x = -1 , g"(x) > 0 so f'(x) = g(x) has minimum value at x = -1 which is g(-1) = 3 - 6 + 3 = 0
Therefore we can easily say that the option (d) is the correct answer
Free Test
FREE
| Start Free Test |
Community Answer
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x...
Answer:
To determine whether the function f(x) = x^3 - 3x^2 + 3x + 2 has a minimum or maximum at x = -1, we need to analyze the behavior of the function in the vicinity of that point.
First Derivative Test:
One way to determine whether a function has a minimum or maximum at a given point is by using the first derivative test. The first derivative of f(x) can be found by differentiating the function with respect to x:
f'(x) = 3x^2 - 6x + 3
Critical Points:
To find the critical points of the function, we set the first derivative equal to zero and solve for x:
3x^2 - 6x + 3 = 0
Dividing both sides by 3 gives:
x^2 - 2x + 1 = 0
Factoring the quadratic equation gives:
(x - 1)^2 = 0
Therefore, x = 1 is the only critical point of the function.
Second Derivative Test:
To determine whether the critical point x = 1 corresponds to a minimum or maximum, we can use the second derivative test. The second derivative of f(x) can be found by differentiating the first derivative with respect to x:
f''(x) = 6x - 6
Substituting x = -1:
To determine the behavior of the function at x = -1, we substitute x = -1 into the second derivative:
f''(-1) = 6(-1) - 6 = -6 - 6 = -12
Since the second derivative is negative at x = -1, the function f(x) has a local maximum at x = -1.
Therefore, the correct answer is option 'D', f(x) has a minimum at x = -1.
To determine whether the function f(x) = x^3 - 3x^2 + 3x + 2 has a minimum or maximum at x = -1, we need to analyze the behavior of the function in the vicinity of that point.
First Derivative Test:
One way to determine whether a function has a minimum or maximum at a given point is by using the first derivative test. The first derivative of f(x) can be found by differentiating the function with respect to x:
f'(x) = 3x^2 - 6x + 3
Critical Points:
To find the critical points of the function, we set the first derivative equal to zero and solve for x:
3x^2 - 6x + 3 = 0
Dividing both sides by 3 gives:
x^2 - 2x + 1 = 0
Factoring the quadratic equation gives:
(x - 1)^2 = 0
Therefore, x = 1 is the only critical point of the function.
Second Derivative Test:
To determine whether the critical point x = 1 corresponds to a minimum or maximum, we can use the second derivative test. The second derivative of f(x) can be found by differentiating the first derivative with respect to x:
f''(x) = 6x - 6
Substituting x = -1:
To determine the behavior of the function at x = -1, we substitute x = -1 into the second derivative:
f''(-1) = 6(-1) - 6 = -6 - 6 = -12
Since the second derivative is negative at x = -1, the function f(x) has a local maximum at x = -1.
Therefore, the correct answer is option 'D', f(x) has a minimum at x = -1.
|
Explore Courses for JEE exam
|
|
Top Courses for JEEView all
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer?
Question Description
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. 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 Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer?.
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. 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 Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer?.
Solutions for Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. 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 Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer?, a detailed solution for Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? has been provided alongside types of Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Let f(x) = x3 + 3x2 + 3x + 2. Then, at x = -1a)f(x) has a maximumb)f(x) has a minimumc)f'(x) has a maximumd)f'(x) has a minimumCorrect answer is option 'D'. Can you explain this answer? tests, examples and also practice JEE tests.
|
|
Explore Courses for JEE exam
|
|
Signup for Free!
Signup to see your scores go up within 7 days! Learn & Practice with 1000+ FREE Notes, Videos & Tests.
|
© EduRev
|
Education Revolution
|
|
Signup to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup