To find the critical numbers of the function

$f(x) = x^{4/5} (x - 2)^7$

we need to take the derivative $f'(x)$ and set it equal to zero. Critical points occur where $f'(x) = 0$ or where $f'(x)$ is undefined.

Step 1: Find $f'(x)$

Let's use the product rule since $f(x) = u(x) \cdot v(x)$, where $u(x) = x^{4/5}$ and $v(x) = (x - 2)^7$.

The product rule states: $f'(x) = u'(x) v(x) + u(x) v'(x)$

Finding $u'(x)$

$u(x) = x^{4/5} \Rightarrow u'(x) = \frac{4}{5} x^{-1/5}$

Finding $v'(x)$

$v(x) = (x - 2)^7 \Rightarrow v'(x) = 7(x - 2)^6$

Now substitute $u(x)$, $u'(x)$, $v(x)$, and $v'(x)$ into the product rule formula:

$f'(x) = \frac{4}{5} x^{-1/5} (x - 2)^7 + x^{4/5} \cdot 7(x - 2)^6$

Step 2: Set $f'(x) = 0$

To find the critical points, factor $f'(x)$ as follows:

$f'(x) = x^{-1/5} (x - 2)^6 \left( \frac{4}{5} (x - 2) + 7x \right)$

Now solve for $x$ in each factor:

1. $x^{-1/5} = 0$: This has no solution because $x^{-1/5}$ is undefined at $x = 0$.

2. $(x - 2)^6 = 0$: This gives $x = 2$.

3. Solve $\frac{4}{5} (x - 2) + 7x = 0$: $\frac{4}{5}x - \frac{8}{5} + 7x = 0$

Solving this, we find an additional critical number.