The problem is about determining the point at which the sequence $f_n(x) = n x (1 - x)^n$ is not uniformly convergent on the interval $[0, 1]$.

Solution:

1. Expression Analysis:
   The sequence is given as:
   $f_n(x) = n x (1 - x)^n$
   The term $(1 - x)^n$ becomes very small as $n \to \infty$, except for values of $x$ near 0 or 1.

2. Uniform Convergence:
   To check uniform convergence, we examine the behavior of $f_n(x)$ on $[0, 1]$. The sequence converges pointwise to 0 for every $x$ in $[0, 1]$, but we need to check if the convergence is uniform.

3. Critical Point Analysis:
   Let us find the maximum of $f_n(x)$:
   $f_n'(x) = n(1 - x)^n - n^2 x (1 - x)^{n-1} = 0$
   Factoring out $n (1 - x)^{n-1}$, we get:
   $1 - n x = 0 \implies x = \frac{1}{n}$
   At $x = \frac{1}{n}$, $f_n(x)$ attains its maximum:
   f_n\left(\frac{1}{n}\right) = n \cdot \frac{1}{n} \cdot \left(1 - \frac{1}{n}\right)^n \approx 1 \cdot e^{-1} \quad \text{(as n \to \infty)}.

4. Behavior at Specific Points:

   • At $x = 0$: $f_n(0) = 0$ for all $n$.
   • At $x = 1$: $f_n(1) = 0$ for all $n$.
   • At $x = \frac{1}{2}$:
     f_n\left(\frac{1}{2}\right) = n \cdot \frac{1}{2} \cdot \left(\frac{1}{2}\right)^n = \frac{n}{2^{n+1}} \to 0 \quad \text{as n \to \infty}.
     Hence, no uniform convergence issue at $x = \frac{1}{2}$.

5. Non-uniform Convergence Point:
   Near $x = \frac{1}{n}$, the convergence of $f_n(x)$ to 0 is slow and the sequence achieves its maximum. As $n \to \infty$, this makes the convergence non-uniform around $x = 0$.

Answer:

The sequence is not uniformly convergent at $x = 0$.
Correct option: (c) $x = 0$.

Would you like a detailed explanation of uniform convergence?

5 Related Questions:

1. What is the difference between pointwise and uniform convergence?
2. How is the derivative used to find the critical points of $f_n(x)$?
3. What is the significance of the term $(1 - x)^n$ as $n \to \infty$?
4. How does uniform convergence depend on the maximum of the sequence?
5. Can a sequence be uniformly convergent on a subset of its domain but not on the entire domain?

Tip:

Uniform convergence can be verified by checking if the maximum deviation $\sup |f_n(x) - f(x)|$ tends to 0 as $n \to \infty$.