Subtopic: Stress and Strain Tensors & Generalized Hooke's Law
Question Statement:
The real function $y = \sin^2(|x|)$ is:
Scientific schematic diagram reproduced from official IIT paper
Options:
(A) continuous for all $x$
(B) differentiable for all $x$
(C) not continuous at $x = 0$
(D) not differentiable at $x = 0$
Official Answer Key:
A, B
Step-by-Step Solution & Explanation
### **Detailed Step-by-Step Solution (GATE AE 2022 Q.43)**
**1. Parity and Differentiability**:
Note that $\sin^2(|x|) = (\sin |x|)^2 = (\pm \sin x)^2 = \sin^2 x$ for all $x \in \mathbb{R}$.
Since $y = \sin^2 x$ is an infinitely differentiable smooth function everywhere on $\mathbb{R}$:
- It is continuous for all $x$ (Option A).
- It is differentiable for all $x$ (Option B).
- **Official Answer Key**: **A, B**
