Which of the following statement(s) is/are true about the function defined as \( f(x)= e^{-x} \lvert \cos x \rvert \) for \( x>0 \)?
Which of the following statement(s) is/are true about the function defined as \( f(x)= e^{-x} \lvert \cos x \rvert \) for \( x>0 \)?
Show Hint
Differentiable at \( x = \dfrac{\pi}{2} \)
- Differentiable at \( x = \pi \)
Differentiable at \( x = \dfrac{3\pi}{2} \)
- Continuous at \( x = 2\pi \)
The Correct Option is B, D
Solution and Explanation
Step 1: Check continuity.
Since \( e^{-x} \) is continuous and \( |\cos x| \) is also continuous everywhere, the product is continuous for all \( x>0 \). This includes \( x = 2\pi \). Thus, (D) is true.
Step 2: Check differentiability.
Non-differentiability occurs where \( |\cos x| \) has a kink. This happens when \( \cos x = 0 \), i.e. at
\[
x = \frac{\pi}{2}, \frac{3\pi}{2}, \dots
\]
Thus, function is not differentiable at \( \frac{\pi}{2} \) and \( \frac{3\pi}{2} \). So (A) and (C) are false.
At \( x = \pi \), \(\cos(\pi) = -1\), and there is no kink. Both \( e^{-x} \) and \( |\cos x| \) are smooth here; hence the product is differentiable. So (B) is true.
Final Answer: (B), (D)
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