Does f_n(x) = nx^n(1-x) converge uniformly on [0,1]?

[Artusartos]

Homework Statement

Consider $$f_n(x) = nx^n(1-x)$$ for x in [0,1].

a) What is the limit of $$f_n(x)$$?

b) Does $$f_n \rightarrow f$$ uniformly on [0,1]?

Homework Equations

The Attempt at a Solution

a) 0

b) Yes...

We know that $$sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|$$...

and

$$lim_{n \rightarrow \infty} [sup\{ |f_n(x) - f(x)|: x \in [0,1]\}] = 0$$

So it must be uniformly convergent on [0,1].

Do you think my answer is correct?Thanks in advance

 

[micromass]

Homework Statement

Consider $$f_n(x) = nx^n(1-x)$$ for x in [0,1].

a) What is the limit of $$f_n(x)$$?

b) Does $$f_n \rightarrow f$$ uniformly on [0,1]?

Homework Equations

The Attempt at a Solution

a) 0

b) Yes...

We know that $$sup|f_n(x) - f(x)| = |n{\frac{1}{2}}^n(1-\frac{1}{2})|$$...

Why? How do you know they will always obtain their maximum in 1/2??

 

[jbunniii]

If we let $y = 1-x$, then we may write
$$f_n(1-y) = n y (1 - y)^n$$
Now what happens if you choose $y = 1/n$?