Can Bernstein Polynomials Approximate Functions and Their Derivatives Uniformly?

[adpc]

Homework Statement

Show that if f is continuously differentiable on [a, b], then there is a sequence of
polynomials pn converging uniformly to f such that p'n converge uniformly to f' as
well.

Homework Equations

The Attempt at a Solution

Let pn(t) = cn t^n
Use uniform convergence and integrate from a to x to get:
$$f(x) - f(a) = \lim_{n \rightarrow \infty} \left[ \frac{c_n}{n+1} ( x^{n+1} - a^{n+1}) \right]$$

Now what is next?? How do I show that f(a) converges to the right side of the limit? so that then i can conclude that f(x) = lim cn x^(n+1) / (n+1)

 

[mighty2000]

= lim p'n(t)

To show that f(a) converges to the right side of the limit, we can use the fact that f is continuously differentiable on [a,b] to apply the Mean Value Theorem. Since f is continuously differentiable, it is also differentiable, which means that f' is continuous on [a,b]. Therefore, by the Mean Value Theorem, there exists a c in (a,x) such that f'(c) = (f(x) - f(a))/(x-a).

Using this, we can rewrite the limit as:

lim_{n \rightarrow \infty} \left[ \frac{c_n}{n+1} ( x^{n+1} - a^{n+1}) \right] = lim_{n \rightarrow \infty} \left[ \frac{c_n}{n+1} ( x-a)(x^n + x^{n-1}a + ... + a^n) \right]

Since c_n is a constant and x-a is a fixed value, we can focus on the term (x^n + x^{n-1}a + ... + a^n) and show that it converges to f'(c) as n approaches infinity.

We can use the fact that f' is continuous on [a,b] to apply the Intermediate Value Theorem. This means that for any value of x in [a,b], there exists a d in (a,x) such that f'(d) = (f(x) - f(a))/(x-a).

Using this, we can rewrite the term (x^n + x^{n-1}a + ... + a^n) as:

(x^n + x^{n-1}a + ... + a^n) = (x^n + x^{n-1}a + ... + a^n) - (a^n + a^{n-1}a + ... + a^n) + (a^n + a^{n-1}a + ... + a^n)

= (x^n - a^n) + (x^{n-1}a - a^{n-1}a) + ... + (x-a)a^n + (a^n + a^{n-1}a + ... + a^n)

= (x-a)(x^{n-1} + x^{n-2}a + ... + a^{