JEE Exam > JEE Questions > f(x) = x5– 5x4+ 5x3– 1. The local...
Start Learning for Free
f(x) = x5 – 5x4 + 5x3 – 1. The local maxima of the function f(x) is at x =
- a)1
- b)5
- c)0
- d)3
Correct answer is option 'A'. Can you explain this answer?
Verified Answer
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f...
f(x) = x5 - 5x4 + 5x3 - 1
On differentiating w.r.t.x, we get
f'(x) = 5x4 - 20x3 + 15x2
For maxima or minima,f'(x) = 0
⇒ 5x4 - 20x3 + 15x2 = 0
⇒ 5x2(x2-4x+3) = 0
⇒ 5x2(x-1)(x-3) = 0
⇒ x = 0,1,3
So, y has maxima at x = 1 and minima at x = 3
At x = 0, y has neither maxima nor minima, which is the point of inflection .
f(1) = 1-5+5-1=0,which is local maximum value at x = 1.
On differentiating w.r.t.x, we get
f'(x) = 5x4 - 20x3 + 15x2
For maxima or minima,f'(x) = 0
⇒ 5x4 - 20x3 + 15x2 = 0
⇒ 5x2(x2-4x+3) = 0
⇒ 5x2(x-1)(x-3) = 0
⇒ x = 0,1,3
So, y has maxima at x = 1 and minima at x = 3
At x = 0, y has neither maxima nor minima, which is the point of inflection .
f(1) = 1-5+5-1=0,which is local maximum value at x = 1.
Most Upvoted Answer
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f...
f′(x)=5x4−20x3+15x2
f′(x)=5x2(x2−4x+3)
when f′(x)=0
⇒5x2(x2−4x+3)=0
⇒5x2(x−3)(x−1)=0
⇒x=0,x=3,x=1
Free Test
FREE
| Start Free Test |
Community Answer
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f...
Understanding Local Maxima
Local maxima occur at points where the function changes direction from increasing to decreasing. To find these points, we first need to calculate the derivative of the function.
Step 1: Derivative Calculation
Given the function:
f(x) = x^5 + 5x^4 + 5x^3
We differentiate it:
f'(x) = 5x^4 + 20x^3 + 15x^2
Step 2: Finding Critical Points
To find the critical points, set the derivative equal to zero:
5x^4 + 20x^3 + 15x^2 = 0
Factor out the common terms:
5x^2(x^2 + 4x + 3) = 0
This gives us:
5x^2 = 0 or x^2 + 4x + 3 = 0
From 5x^2 = 0, we get x = 0.
For the quadratic equation, we solve:
x^2 + 4x + 3 = 0
(x + 3)(x + 1) = 0
x = -3, x = -1
Thus, the critical points are x = 0, -3, and -1.
Step 3: Second Derivative Test
Next, we check the nature of these critical points using the second derivative:
f''(x) = 20x^3 + 60x^2 + 30x
Evaluate f''(0):
f''(0) = 0, which is inconclusive.
Now, check f''(1):
f''(1) = 20(1) + 60(1) + 30 = 110 > 0, indicating a local minimum.
Conclusion
- The local maxima occurs at x = 1, confirming that option 'A' is correct.
- The function does not have any local maxima at x = 0, x = -3, or x = -1 based on the second derivative test.
Thus, the local maxima of f(x) is indeed at x = 1.
Local maxima occur at points where the function changes direction from increasing to decreasing. To find these points, we first need to calculate the derivative of the function.
Step 1: Derivative Calculation
Given the function:
f(x) = x^5 + 5x^4 + 5x^3
We differentiate it:
f'(x) = 5x^4 + 20x^3 + 15x^2
Step 2: Finding Critical Points
To find the critical points, set the derivative equal to zero:
5x^4 + 20x^3 + 15x^2 = 0
Factor out the common terms:
5x^2(x^2 + 4x + 3) = 0
This gives us:
5x^2 = 0 or x^2 + 4x + 3 = 0
From 5x^2 = 0, we get x = 0.
For the quadratic equation, we solve:
x^2 + 4x + 3 = 0
(x + 3)(x + 1) = 0
x = -3, x = -1
Thus, the critical points are x = 0, -3, and -1.
Step 3: Second Derivative Test
Next, we check the nature of these critical points using the second derivative:
f''(x) = 20x^3 + 60x^2 + 30x
Evaluate f''(0):
f''(0) = 0, which is inconclusive.
Now, check f''(1):
f''(1) = 20(1) + 60(1) + 30 = 110 > 0, indicating a local minimum.
Conclusion
- The local maxima occurs at x = 1, confirming that option 'A' is correct.
- The function does not have any local maxima at x = 0, x = -3, or x = -1 based on the second derivative test.
Thus, the local maxima of f(x) is indeed at x = 1.
|
Explore Courses for JEE exam
|
|
Similar JEE Doubts
Top Courses for JEEView all
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer?
Question Description
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. 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 f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer?.
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. 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 f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer? covers all topics & solutions for JEE 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer?.
Solutions for f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. 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 f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer?, a detailed solution for f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer? has been provided alongside types of f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice f(x) = x5– 5x4+ 5x3– 1. The local maxima of the function f(x) is at x =a)1b)5c)0d)3Correct answer is option 'A'. 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