2. Higher order derivatives Find $$ f^{\prime \prime \prime}(x) $$ if $$ f(x)=6 x^{5}-4 x^{2} $$. 3. Graphing Where does the graph of $$ y=\frac{x^{2}+9}{x} $$ have a horizontal tangent line?

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**Problem 2. Higher Order Derivatives**

Given $$ f(x) = 6x^5 - 4x^2 $$, we want to find $$ f^{\prime\prime\prime}(x) $$.

1. Compute the first derivative:
   $$
   f'(x) = \frac{d}{dx}(6x^5) - \frac{d}{dx}(4x^2) = 30x^4 - 8x.
   $$

2. Compute the second derivative:
   $$
   f''(x) = \frac{d}{dx}(30x^4) - \frac{d}{dx}(8x) = 120x^3 - 8.
   $$

3. Compute the third derivative:
   $$
   f'''(x) = \frac{d}{dx}(120x^3) - \frac{d}{dx}(8) = 360x^2.
   $$

Thus, 
$$
f^{\prime\prime\prime}(x) = 360x^2.
$$

---

**Problem 3. Graphing**

We are given the function 
$$
y = \frac{x^2 + 9}{x}.
$$
First, rewrite the function by splitting the fraction:
$$
y = \frac{x^2}{x} + \frac{9}{x} = x + \frac{9}{x}.
$$

A horizontal tangent line occurs where the derivative $$ y' $$ is zero.

1. Differentiate $$ y $$ with respect to $$ x $$:
   $$
   y' = \frac{d}{dx}\left(x + \frac{9}{x}\right) = 1 - \frac{9}{x^2}.
   $$

2. Set the derivative equal to zero to find the critical points:
   $$
   1 - \frac{9}{x^2} = 0.
   $$
   Solve for $$ x $$:
   $$
   \frac{9}{x^2} = 1 \quad \Longrightarrow \quad x^2 = 9 \quad \Longrightarrow \quad x = 3 \quad \text{or} \quad x = -3.
   $$

3. Find the corresponding $$ y $$-values:

- For $$ x = 3 $$:
     $$
     y = 3 + \frac{9}{3} = 3 + 3 = 6.
     $$
   - For $$ x = -3 $$:
     $$
     y = -3 + \frac{9}{-3} = -3 - 3 = -6.
     $$

Thus, the horizontal tangent lines occur at the points:
$$
(3, 6) \quad \text{and} \quad (-3, -6).
$$
**Problem 2. Higher Order Derivatives** Given $$ f(x) = 6x^5 - 4x^2 $$, the third derivative $$ f^{\prime\prime\prime}(x) $$ is: $$ f^{\prime\prime\prime}(x) = 360x^2. $$ --- **Problem 3. Graphing** The function $$ y = \frac{x^2 + 9}{x} $$ has horizontal tangent lines at the points: $$ (3, 6) \quad \text{and} \quad (-3, -6). $$

2. Higher order derivatives Find \( f^{\prime \prime \prime}(x) \) if \( f(x)=6 x^{5}-4 x^{2} \). 3. Graphing Where does the graph of \( y=\frac{x^{2}+9}{x} \) have a horizontal tangent line?

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