Differential Equations
Product Rule
MJAT None
Grade 12
Question:
If $f(x) = x^3 - x^2 + (x-1) sin x$ and $g(x)$ is an arbitrary function, then which of the following statements is/are TRUE?
A) If $g$ is continuous at $x= 1$, then $fg$ is differentiable at $x= 1$
B) If $fg$ is differentiable at $x= 1$, then $g$ is continuous at $x= 1$
C) If $g$ is differentiable at $x= 1$, then $fg$ is differentiable at $x= 1$
D) If $fg$ is differentiable at $x= 1$, then $g$ is differentiable at $x= 1$
Step-by-Step Solution
Key Concept: Rolle’s Theorem guarantees a critical point in $ (0,1) $ due to $ f(0) = f(1) = 0 $, and the cubic term dominates the function’s end behavior.
1. Evaluate $ f(0) $ and $ f(1) $:
$ f(0) = 0^3 - 0^2 + (-1)\sin(0) = 0 $,
$ f(1) = 1^3 - 1^2 + 0 \cdot \sin(1) = 0 $.
2. Apply Rolle’s Theorem: Since $ f(x) $ is continuous and differentiable on $[0,1]$ and $ f(0) = f(1) = 0 $, there exists $ c \in (0,1) $ such that $ f'(c) = 0 $.
3. Compute $ f'(x) $:
$ f'(x) = 3x^2 - 2x + \sin x + (x - 1)\cos x $.
The derivative combines polynomial and trigonometric terms, but its sign cannot be determined globally without further analysis.
4. Analyze behavior as $ x \to \infty $:
The dominant term $ x^3 $ ensures $ f(x) \to \infty $, while $ \sin x $ oscillates but does not affect the asymptotic trend.
Correct Answer: A, C