Differentiability
Product Rule
MMTS_Full_Test_18
Grade 12
Question:
Let $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=x^3-x^2+(x-1)\sin x$ and $g:\mathbb{R}\to\mathbb{R}$ be arbitrary. Let $fg$ be the product function. Number of correct statements: A) If $g$ is discontinuous at $x=1$, then $fg$ can never be differentiable at $x=1$. B) If $fg$ is differentiable at $x=1$, then $g$ is continuous at $x=1$. C) If $fg$ is differentiable at $x=1$, then $g$ must be differentiable at $x=1$.
Step-by-Step Solution
Key Concept: $f(1)=0$ and $f'(1)=0$; if $f(1)=0$, differentiability of $fg$ at $x=1$ doesn't require continuity of $g$
$f(1)=0$. A) False: $fg$ can be differentiable even if $g$ is discontinuous at $x=1$ (since $f(1)=0$). B) False: $g$ need not be continuous. C) False: $g$ need not be differentiable. 0 correct statements.
Correct Answer: 4