Question

For the function \( f(x) = e^x |\sin x|, \; x \in \mathbb{R}, \) which of the following statements is/are TRUE?}

• A. The function is continuous at all \(x\)
• B. The function is differentiable at all \(x\)
• C. The function is periodic
• D. The function is bounded

Hint:

Remember: If a function is a product of continuous functions, it is continuous. But differentiability fails where absolute value functions have sharp corners (like at multiples of \(\pi\) here).

Answer 1

The Correct Option is A

Step 1: Analyze continuity.
The function is a product of two functions: \(e^x\) (continuous for all \(x \in \mathbb{R}\)) and \(|\sin x|\) (continuous for all \(x \in \mathbb{R}\)).
Hence, their product \(f(x) = e^x |\sin x|\) is continuous for all real \(x\).

Step 2: Analyze differentiability.
\(|\sin x|\) is not differentiable at points where \(\sin x = 0\) (i.e., at \(x = n\pi, \; n \in \mathbb{Z}\)).
Thus, \(f(x)\) is not differentiable at these points. Hence (B) is false.

Step 3: Analyze periodicity.
\(e^x\) is not periodic. Since \(f(x)\) includes \(e^x\), the function cannot be periodic. Hence (C) is false.

Step 4: Analyze boundedness.
As \(x \to \infty\), \(e^x |\sin x| \to \infty\). Hence, the function is unbounded. So (D) is false.
Therefore, the only true statement is (A).
\[ \boxed{\text{The function is continuous at all } x} \]