Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) $$ F(x)=x^{4 / 5}(x-9)^{2} $$ $$ x=\square $$

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We are given
$$
F(x)=x^{\frac{4}{5}}(x-9)^2.
$$
The first step is to find its derivative $$ F'(x) $$.

**Step 1. Compute the derivative using the product rule**

Let
$$
u(x) = x^{\frac{4}{5}} \quad \text{and} \quad v(x)=(x-9)^2.
$$
Then,
$$
u'(x)=\frac{4}{5}x^{\frac{4}{5}-1}=\frac{4}{5}x^{-\frac{1}{5}},
$$
and
$$
v'(x)=2(x-9).
$$
Using the product rule,
$$
F'(x)=u'(x)v(x)+u(x)v'(x)=\frac{4}{5}x^{-\frac{1}{5}}(x-9)^2 + x^{\frac{4}{5}}\cdot 2(x-9).
$$

**Step 2. Factor common terms in the derivative**

Notice that both terms share a factor of $$ x^{-\frac{1}{5}} $$ and $$ (x-9) $$. We can factor these out:
$$
F'(x)=x^{-\frac{1}{5}}(x-9)\left[\frac{4}{5}(x-9)+2x\right].
$$

**Step 3. Set the derivative equal to zero or determine where it is undefined**

The derivative $$ F'(x) $$ is zero when any of the factors equals zero or it is undefined while $$ F(x) $$ is defined.

The factors are:
1. $$ x^{-\frac{1}{5}} $$
2. $$ (x-9) $$
3. $$ \frac{4}{5}(x-9)+2x $$

- **Factor 1: $$ x^{-\frac{1}{5}} $$**
  
  This expression is undefined when $$ x=0 $$ since negative exponents imply division by $$ x^{\frac{1}{5}} $$. Since $$ F(0) $$ is defined (because the exponent $$ \frac{4}{5} $$ is acceptable for $$ x=0 $$), $$ x=0 $$ is a critical number due to the derivative being undefined there.

- **Factor 2: $$ (x-9) $$**
  
  Setting
  $$
  x-9=0 \quad \Rightarrow \quad x=9.
  $$

- **Factor 3: $$ \frac{4}{5}(x-9)+2x $$**
  
  Set the factor equal to zero:
  $$
  \frac{4}{5}(x-9) + 2x = 0.
  $$
  Multiply both sides by 5 to eliminate the fraction:
  $$
  4(x-9) + 10x = 0.
  $$
  Simplify:
  $$
  4x - 36 + 10x = 0 \quad \Rightarrow \quad 14x - 36 = 0.
  $$
  Solve for $$ x $$:
  $$
  14x=36 \quad \Rightarrow \quad x=\frac{36}{14}=\frac{18}{7}.
  $$

**Step 4. List all critical numbers**

The three critical numbers are:
$$
x=0,\quad x=\frac{18}{7},\quad x=9.
$$

Thus, the critical numbers of the function are:

$$0, \frac{18}{7}, 9$$.
The critical numbers are $$0$$, $$\frac{18}{7}$$, and $$9$$.

Find the critical numbers of the function. (Enter your answers as a comma-separated list. If an answer does not exist, enter DNE.) \( F(x)=x^{4 / 5}(x-9)^{2} \) \( x=\square \)

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