Find the Taylor polynomial p3(x) for f(x)=5x1/3 at x=3 .

The Taylor polynomial of degree n for a function f(x) at x=a is given by:

Pn(x) = f(a) + f'(a)(x-a) + f''(a)(x-a)^2/2! + ... + f^n(a)(x-a)^n/n!

Here, we want to find the Taylor polynomial of degree 3, P3(x), for f(x) = 5x^(1/3) at x=3.

First, we need to find the first, second, and third derivatives of f(x).

f'(x) = 5/3 * x^(-2/3)

f''(x) = -10/9 * x^(-5/3)

f'''(x) = 50/27 * x^(-8/3)

Now, we evaluate these at x=3:

f(3) = 53^(1/3) = 53^(1/3)

f'(3) = 5/3 * 3^(-2/3) = 5/3 * 3^(-2/3)

f''(3) = -10/9 * 3^(-5/3) = -10/9 * 3^(-5/3)

f'''(3) = 50/27 * 3^(-8/3) = 50/27 * 3^(-8/3)

Finally, we substitute these values into the Taylor polynomial formula:

P3(x) = f(3) + f'(3)(x-3) + f''(3)(x-3)^2/2! + f'''(3)(x-3)^3/3!

This is the Taylor polynomial of degree 3 for f(x) = 5x^(1/3) at x=3.