Differentiability
Statement Type
MMTS_Full_Test_01
Grade 12
Question:
Statement-I: $f:\mathbb{R}\to\mathbb{R}$ be a function such that $|f(x)|\le x^2$, $\forall x\in\mathbb{R}$, then $f(x)$ is differentiable at $x=0$. Statement-II: $f:\mathbb{R}\to\mathbb{R}$ such that $|f(x)|\le x^p$, $\forall x\in\mathbb{R}$, then $f(x)$ is differentiable at $x=0$ for any whole number $p$.
Both S-I and S-II are true
Both S-I and S-II are false
S-I is true and S-II is false
S-I is false and S-II is true
Step-by-Step Solution
Key Concept: S-I: squeeze theorem shows $f'(0)=0$. S-II: fails for $p=0$ or $p=1$
S-I true (squeeze). S-II false for $p=1$ ($f(x)=|x|$ satisfies $|f|\le|x|$ but not differentiable at 0). Answer: S-I true, S-II false.
Correct Answer: 3