Mechanical Engineering Exam > Mechanical Engineering Questions > Consider the function f(x) = |x|3, where x is...
Start Learning for Free
Consider the function f(x) = |x|3, where x is real
. Then the function f(x) at x = 0 is
a)Continuous but not differentiable
b)Once differentiable but not twice
c)Twice differentiable but not thrice
d)Thrice differentiable
Correct answer is option 'C
'. Can you explain this answer?| FREE This question is part of | Download PDF Attempt this Test |
Verified Answer
Consider the function f(x) = |x|3, where x is real... more. Then the f...
Most Upvoted Answer
Consider the function f(x) = |x|3, where x is real... more. Then the f...
Free Test
FREE
| Start Free Test |
Community Answer
Consider the function f(x) = |x|3, where x is real... more. Then the f...
Function Analysis:
The given function is f(x) = |x|^3, where x is a real number. To analyze the function, we need to consider the cases when x is positive and when x is negative.
Case 1: x > 0
When x is positive, the absolute value of x is simply x. Therefore, the function becomes f(x) = x^3, which is a polynomial function. Polynomials are continuous and differentiable for all real numbers.
Case 2: x < />
When x is negative, the absolute value of x is -x. Therefore, the function becomes f(x) = (-x)^3 = -x^3. This is also a polynomial function.
Function Analysis (continued):
From the above analysis, we can conclude that the function f(x) = |x|^3 is a polynomial function, which means it is both continuous and differentiable for all real numbers.
Differentiability at x = 0:
To determine the differentiability of the function at x = 0, we need to calculate the left-hand and right-hand limits of the derivative of f(x) as x approaches 0.
Left-hand limit:
lim┬(h→0-)〖(f(0+h)-f(0))/h〗 = lim┬(h→0-)〖(f(h)-f(0))/h〗〗
= lim┬(h→0-)〖((|h|^3-0^3)/h)〗
= lim┬(h→0-)(|h|^2)
= 0^2
= 0
Right-hand limit:
lim┬(h→0+)〖(f(0+h)-f(0))/h〗 = lim┬(h→0+)〖(f(h)-f(0))/h〗〗
= lim┬(h→0+)〖((|h|^3-0^3)/h)〗
= lim┬(h→0+)(|h|^2)
= 0^2
= 0
Conclusion:
The left-hand and right-hand limits of the derivative of f(x) as x approaches 0 both exist and are equal to 0. Therefore, the derivative of f(x) exists at x = 0, which implies that the function is differentiable at x = 0.
Final Answer:
The function f(x) = |x|^3 is continuous and differentiable at x = 0. Therefore, the correct answer is option 'C' - Twice differentiable but not thrice.
The given function is f(x) = |x|^3, where x is a real number. To analyze the function, we need to consider the cases when x is positive and when x is negative.
Case 1: x > 0
When x is positive, the absolute value of x is simply x. Therefore, the function becomes f(x) = x^3, which is a polynomial function. Polynomials are continuous and differentiable for all real numbers.
Case 2: x < />
When x is negative, the absolute value of x is -x. Therefore, the function becomes f(x) = (-x)^3 = -x^3. This is also a polynomial function.
Function Analysis (continued):
From the above analysis, we can conclude that the function f(x) = |x|^3 is a polynomial function, which means it is both continuous and differentiable for all real numbers.
Differentiability at x = 0:
To determine the differentiability of the function at x = 0, we need to calculate the left-hand and right-hand limits of the derivative of f(x) as x approaches 0.
Left-hand limit:
lim┬(h→0-)〖(f(0+h)-f(0))/h〗 = lim┬(h→0-)〖(f(h)-f(0))/h〗〗
= lim┬(h→0-)〖((|h|^3-0^3)/h)〗
= lim┬(h→0-)(|h|^2)
= 0^2
= 0
Right-hand limit:
lim┬(h→0+)〖(f(0+h)-f(0))/h〗 = lim┬(h→0+)〖(f(h)-f(0))/h〗〗
= lim┬(h→0+)〖((|h|^3-0^3)/h)〗
= lim┬(h→0+)(|h|^2)
= 0^2
= 0
Conclusion:
The left-hand and right-hand limits of the derivative of f(x) as x approaches 0 both exist and are equal to 0. Therefore, the derivative of f(x) exists at x = 0, which implies that the function is differentiable at x = 0.
Final Answer:
The function f(x) = |x|^3 is continuous and differentiable at x = 0. Therefore, the correct answer is option 'C' - Twice differentiable but not thrice.
Attention Mechanical Engineering Students!
To make sure you are not studying endlessly, EduRev has designed Mechanical Engineering study material, with Structured Courses, Videos, & Test Series. Plus get personalized analysis, doubt solving and improvement plans to achieve a great score in Mechanical Engineering.
|
Explore Courses for Mechanical Engineering exam
|
|
Similar Mechanical Engineering Doubts
Top Courses for Mechanical EngineeringView all
Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer?
Question Description
Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer?.
Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? for Mechanical Engineering 2024 is part of Mechanical Engineering preparation. The Question and answers have been prepared according to the Mechanical Engineering exam syllabus. Information about Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Mechanical Engineering 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer?.
Solutions for Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? in English & in Hindi are available as part of our courses for Mechanical Engineering.
Download more important topics, notes, lectures and mock test series for Mechanical Engineering Exam by signing up for free.
Here you can find the meaning of Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? defined & explained in the simplest way possible. Besides giving the explanation of
Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer?, a detailed solution for Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? has been provided alongside types of Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? theory, EduRev gives you an
ample number of questions to practice Consider the function f(x) = |x|3, where x is real... more. Then the function f(x) at x = 0 isa)Continuous but not differentiableb)Once differentiable but not twicec)Twice differentiable but not thriced)Thrice differentiableCorrect answer is option 'C'. Can you explain this answer? tests, examples and also practice Mechanical Engineering tests.
|
|
Explore Courses for Mechanical Engineering 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 to see your scores
go up within 7 days!
Access 1000+ FREE Docs, Videos and Tests
Takes less than 10 seconds to signup