JEE Mains · Maths · STD 12 - 5. continuity and differentiation
For the function \(f(x) = e^{\sin|x|} - |x|\), \(x \in \mathbb{R}\), consider the following statements:
Statement I: \(f\) is differentiable for all \(x \in \mathbb{R}\).
Statement II: \(f\) is increasing in \(\left(-\pi, -\dfrac{\pi}{2}\right)\).
In the light of the above statements, choose the correct answer from the options given below:
- A Both Statement I and Statement II are true
- B Both Statement I and Statement II are false
- C Statement I is true but Statement II is false
- D Statement I is false but Statement II is true
Answer & Solution
Correct Answer
(A) Both Statement I and Statement II are true
Step-by-step Solution
Detailed explanation
For Statement I: The given function is \(f(x) = e^{\sin|x|} - |x|\). For \(x > 0\), \(f(x) = e^{\sin x} - x\). Differentiating with respect to \(x\), we get: \(f'(x) = e^{\sin x}\cos x - 1\) The right-hand derivative at \(x = 0\) is:…
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