Let 
f(x) = 10x + (5/x) find increasing and decreasing intervals 
.

To determine the increasing and decreasing intervals of the function \( f(x) = 10x + \frac{5}{x} \), we need to follow these steps:

### Step 1: Find the derivative of \( f(x) \)
The derivative \( f'(x) \) will help us determine where the function is increasing or decreasing.

\[
f'(x) = \frac{d}{dx} \left( 10x + \frac{5}{x} \right)
\]

Using basic differentiation rules:
- The derivative of \( 10x \) is 10.
- The derivative of \( \frac{5}{x} \) is \( -\frac{5}{x^2} \).

So,
\[
f'(x) = 10 - \frac{5}{x^2}
\]

### Step 2: Set \( f'(x) = 0 \) and solve for \( x \)
To find critical points, we set the derivative equal to zero and solve for \( x \):

\[
10 - \frac{5}{x^2} = 0
\]

\[
10 = \frac{5}{x^2}
\]

Multiply both sides by \( x^2 \):

\[
10x^2 = 5
\]

Solve for \( x^2 \):

\[
x^2 = \frac{5}{10} = \frac{1}{2}
\]

Taking the square root of both sides:

\[
x = \pm \frac{1}{\sqrt{2}} = \pm \frac{\sqrt{2}}{2}
\]

### Step 3: Determine the sign of \( f'(x) \)
We need to check the sign of \( f'(x) \) in the intervals determined by the critical points \( x = -\frac{\sqrt{2}}{2} \) and \( x = \frac{\sqrt{2}}{2} \).

- For \( x < -\frac{\sqrt{2}}{2} \), choose a test point (e.g., \( x = -1 \)):
  \[
  f'(-1) = 10 - \frac{5}{(-1)^2} = 10 - 5 = 5 \quad (\text{positive, so } f(x) \text{ is increasing here}).
  \]

- For \( -\frac{\sqrt{2}}{2} < x < \frac{\sqrt{2}}{2} \), choose a test point (e.g., \( x = 0.1 \)):
  \[
  f'(0.1) = 10 - \frac{5}{(0.1)^2} = 10 - 500 = -490 \quad (\text{negative, so } f(x) \text{ is decreasing here}).
  \]

- For \( x > \frac{\sqrt{2}}{2} \), choose a test point (e.g., \( x = 1 \)):
  \[
  f'(1) = 10 - \frac{5}{1^2} = 10 - 5 = 5 \quad (\text{positive, so } f(x) \text{ is increasing here}).
  \]

### Step 4: Conclusion
The function is:
- Increasing on the intervals \( (-\infty, -\frac{\sqrt{2}}{2}) \) and \( (\frac{\sqrt{2}}{2}, \infty) \).
- Decreasing on the interval \( (-\frac{\sqrt{2}}{2}, \frac{\sqrt{2}}{2}) \).

Would you like further details on this solution, or do you have any questions?

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Here are 5 related questions for further practice:
1. How would you find the concavity of the function \( f(x) = 10x + \frac{5}{x} \)?
2. What are the local maxima and minima of \( f(x) \)?
3. Can you sketch the graph of the function based on the information obtained?
4. What would happen to the increasing and decreasing intervals if the function were \( f(x) = ax + \frac{b}{x} \)?
5. How would you apply the First Derivative Test to confirm local extrema?

**Tip:** Always check the sign of the derivative in different intervals around critical points to correctly identify increasing and decreasing behavior.

Let f(x) = 10x + (5/x), find increasing and decreasing intervals.

Discover how to find the increasing and decreasing intervals of the function f(x) = 10x + (5/x) using calculus. This problem involves differentiation, solving for critical points, and applying the First Derivative Test. Suitable for students in Grades 11-12 or early college.

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