A graphing calculator is recommended. Use a graph to find approximate $$ x $$-coordinates of the points of intersection of the answer to two decimal places.) $$ y=4 x \sin \left(x^{2}\right), \quad y=4 x^{4}, x \geq 0 $$

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To find the approximate $$ x $$-coordinates of the points of intersection for the functions 
$$ y = 4x \sin\left(x^2\right) $$
and 
$$ y = 4x^4 $$
with $$ x \geq 0 $$, we can follow these steps:

### 1. Set the Equations Equal
Set the two functions equal to each other to find the points of intersection:
$$
4x \sin\left(x^2\right) = 4x^4
$$
Simplify by dividing both sides by $$ 4x $$ (for $$ x > 0 $$):
$$
\sin\left(x^2\right) = x^3
$$

### 2. Analyze the Equation
We need to solve:
$$
\sin\left(x^2\right) - x^3 = 0
$$
Consider the behavior of both sides:

- At $$ x = 0 $$:
  $$
  \sin(0) = 0 \quad \text{and} \quad 0^3 = 0 \implies (0, 0) \text{ is an intersection point}
  $$
  
- For $$ 0 < x < 1 $$:
  - $$ \sin(x^2) $$ increases initially but will eventually decrease as $$ x $$ approaches 1.
  - $$ x^3 $$ increases steadily.
  
- At $$ x = 1 $$:
  $$
  \sin(1) \approx 0.8415 < 1 = 1^3 \implies \sin(x^2) - x^3 < 0
  $$
  
This indicates there is another intersection point between $$ x = 0 $$ and $$ x = 1 $$.

### 3. Approximate the Second Intersection

Using a graphing approach or numerical methods:

- **At $$ x = 0.9 $$:**
  $$
  \sin(0.81) \approx 0.7295 \quad \text{and} \quad 0.9^3 = 0.729 \implies \sin(x^2) - x^3 \approx 0.0005
  $$
  
- **At $$ x = 0.91 $$:**
  $$
  \sin(0.8281) \approx 0.7373 \quad \text{and} \quad 0.91^3 \approx 0.7536 \implies \sin(x^2) - x^3 \approx -0.0163
  $$
  
The sign changes between $$ x = 0.9 $$ and $$ x = 0.91 $$, indicating an intersection point very close to $$ x = 0.90 $$.

### 4. Conclusion

The approximate $$ x $$-coordinates of the points of intersection, rounded to two decimal places, are:

$$
\boxed{\begin{aligned}
& x = 0.00 \
& x = 0.90
\end{aligned}}
$$

These correspond to the points where the graphs of $$ y = 4x \sin(x^2) $$ and $$ y = 4x^4 $$ intersect for $$ x \geq 0 $$.
The approximate $$ x $$-coordinates of the points of intersection are $$ x = 0.00 $$ and $$ x = 0.90 $$.

A graphing calculator is recommended. Use a graph to find approximate \( x \)-coordinates of the points of intersection of the answer to two decimal places.) \[ y=4 x \sin \left(x^{2}\right), \quad y=4 x^{4}, x \geq 0 \]

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