Functions
Non-differentiability of composite involving optimal f — paragraph I
MJAT_TS4_P2
Grade 12
Question:
**Paragraph I (continued):**
If the maximum value of $L$ is achieved for $f(x)=g(x)$, then the number of points of non-differentiability of $\left[(x^2-1)\left(x-\dfrac{3}{2}\right)g(x)\right]^{2023}$ for all $x\in(0,1)$ is:
Step-by-Step Solution
Key Concept: At maximum: $g(x)=\sqrt{4043}\cdot x^{2021}$. Then $(x^2-1)(x-3/2)g(x)=\sqrt{4043}\cdot x^{2021}(x^2-1)(x-3/2)$, which is a smooth polynomial on $(0,1)$. Raising to odd power $2023$ preserves smoothness.
Zero non-differentiability points on $(0,1)$. Answer: $\mathbf{0}$.
Correct Answer: 0