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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?
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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.
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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 2025 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 2025 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?.
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