Given the function \( f(x) = 7x \sin(x) \), let's analyze it in different aspects:

### 1. Derivative of \( f(x) \):
To find \( f'(x) \), we'll use the **product rule** of differentiation, which states that if \( f(x) = u(x) \cdot v(x) \), then:

\[
f'(x) = u'(x)v(x) + u(x)v'(x)
\]

Here, \( u(x) = 7x \) and \( v(x) = \sin(x) \). Now we differentiate both terms:

\[
u'(x) = 7 \quad \text{and} \quad v'(x) = \cos(x)
\]

Using the product rule:

\[
f'(x) = (7)(\sin(x)) + (7x)(\cos(x)) = 7\sin(x) + 7x\cos(x)
\]

Thus, the derivative is:

\[
f'(x) = 7\sin(x) + 7x\cos(x)
\]

### 2. Critical Points:
To find the critical points, set \( f'(x) = 0 \):

\[
7\sin(x) + 7x\cos(x) = 0
\]

This equation can be simplified to:

\[
\sin(x) + x\cos(x) = 0
\]

This equation may require numerical methods or graphical analysis to solve, depending on the range of \( x \) values you're interested in.

### 3. Integration of \( f(x) \):
To find the integral of \( f(x) = 7x \sin(x) \), we'll use **integration by parts**:

\[
\int 7x \sin(x) \, dx
\]

Let \( u = 7x \) and \( dv = \sin(x) \, dx \), so that \( du = 7 \, dx \) and \( v = -\cos(x) \). Applying the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[
\int 7x \sin(x) \, dx = -7x \cos(x) + 7 \int \cos(x) \, dx
\]

The integral of \( \cos(x) \) is \( \sin(x) \), so the result is:

\[
\int 7x \sin(x) \, dx = -7x \cos(x) + 7\sin(x) + C
\]

### 4. Limits:
The limits of \( f(x) = 7x \sin(x) \) as \( x \to 0 \) and \( x \to \infty \):

- **As \( x \to 0 \)**:
  \[
  f(0) = 7(0)\sin(0) = 0
  \]
  
- **As \( x \to \infty \)**:
  Since \( \sin(x) \) oscillates between -1 and 1, the function doesn't have a finite limit as \( x \to \infty \); instead, it oscillates indefinitely.

### 5. Graphical Behavior:
The function \( f(x) = 7x \sin(x) \) oscillates, and the amplitude of the oscillations grows as \( x \) increases due to the \( 7x \) factor.

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Do you want further details or have any questions about these analyses?

Here are 5 related questions you might explore next:
1. What are the numerical solutions to the equation \( \sin(x) + x\cos(x) = 0 \)?
2. How do we solve \( 7x \sin(x) = k \) for different values of \( k \)?
3. How does the second derivative \( f''(x) \) behave, and what does it tell us about concavity?
4. What is the behavior of the function when restricted to specific intervals like \( [0, \pi] \)?
5. How can we apply this function to model oscillatory motion with increasing amplitude?

**Tip:** Use symmetry properties of trigonometric functions when solving integrals or limits involving \( \sin(x) \) or \( \cos(x) \).

Explore the derivative and integral of f(x) = 7x * sin(x), using calculus principles such as the product rule and integration by parts. This guide also covers finding critical points, limits, and the oscillatory behavior of the function.

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