JEE Exam > JEE Questions > Let f (x) = x | x | and g (x) = sin x.Stateme...
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Let f (x) = x | x | and g (x) = sin x.
Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point.
Statement-2 : gof is twice differentiable at x = 0.
Statement-2 : gof is twice differentiable at x = 0.
- a)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.
- b)Statement-1 is true, Statement-2 is false.
- c)Statement-1 is false, Statement-2 is true.
- d)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentia...
Given that f (x) = x | x | and g (x) = sin x
So that go f (x) = g (f (x)) = g (x | x |) = sin x | x |
So that go f (x) = g (f (x)) = g (x | x |) = sin x | x |
Here we observe L (go f )' (0) = 0 = R (go f)' (0)
⇒ go f is differentiable at x = 0 and (go f)' is continuous at x = 0
∴ L(go f)'' (0) ≠ R (go f )'' (0)
⇒ go f (x) is not twice differentiable at x = 0.
∴ Statement - 1 is true but statement -2 is false.
⇒ go f is differentiable at x = 0 and (go f)' is continuous at x = 0
∴ L(go f)'' (0) ≠ R (go f )'' (0)
⇒ go f (x) is not twice differentiable at x = 0.
∴ Statement - 1 is true but statement -2 is false.
Most Upvoted Answer
Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentia...
Explanation:
Statement-1:
- The composition gof is given by (gof)(x) = g(f(x)) = g(x|x|) = sin(x|x|).
- To check the differentiability of gof at x = 0, we need to calculate the limit as x approaches 0 of [gof(x) - gof(0)] / (x - 0).
- Limit as x approaches 0 of sin(x|x|) = 0.
- Therefore, gof is differentiable at x = 0.
- However, the derivative of sin(x|x|) is not continuous at x = 0, as sin(x) is not differentiable at x = 0. Hence, Statement-1 is false.
Statement-2:
- To check if gof is twice differentiable at x = 0, we need to calculate the second derivative of gof.
- The first derivative of sin(x|x|) is cos(x|x|) * (1 + |x|).
- The second derivative of sin(x|x|) is given by [-sin(x|x|) + 2cos(x|x|) * |x|] * (1 + |x|) + cos(x|x|).
- The second derivative is not continuous at x = 0 as sin(x) and cos(x) are not continuous at x = 0.
- Hence, Statement-2 is false.
Therefore, the correct answer is option B: Statement-1 is true, Statement-2 is false.
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Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point.Statement-2 : gof is twice differentiable at x = 0.a)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.b)Statement-1 is true, Statement-2 is false.c)Statement-1 is false, Statement-2 is true.d)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.Correct answer is option 'B'. Can you explain this answer?
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Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point.Statement-2 : gof is twice differentiable at x = 0.a)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.b)Statement-1 is true, Statement-2 is false.c)Statement-1 is false, Statement-2 is true.d)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.Correct answer is option 'B'. 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 Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point.Statement-2 : gof is twice differentiable at x = 0.a)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.b)Statement-1 is true, Statement-2 is false.c)Statement-1 is false, Statement-2 is true.d)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.Correct answer is option 'B'. Can you explain this answer? covers all topics & solutions for JEE 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let f (x) = x | x | and g (x) = sin x.Statement-1 : gof is differentiable at x = 0 and its derivative is continuous at that point.Statement-2 : gof is twice differentiable at x = 0.a)Statement-1 is true, Statement-2 is true; Statement-2 is not a correct explanation for Statement-1.b)Statement-1 is true, Statement-2 is false.c)Statement-1 is false, Statement-2 is true.d)Statement-1 is true, Statement-2 is true; Statement-2 is a correct explanation for Statement-1.Correct answer is option 'B'. Can you explain this answer?.
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