Using R(x) = 10,000- x^3 + 42x^2 + 600x, 0 ≤ x ≤ 20.

Let's find the inflection point for the function \( R(x) = 10000 - x^3 + 42x^2 + 600x \) over the interval \( 0 \leq x \leq 20 \).

1. **First derivative**:
   \[
   R'(x) = \frac{d}{dx}(10000 - x^3 + 42x^2 + 600x) = -3x^2 + 84x + 600
   \]

2. **Second derivative**:
   \[
   R''(x) = \frac{d}{dx}(-3x^2 + 84x + 600) = -6x + 84
   \]

The inflection point occurs where \( R''(x) = 0 \).

3. **Set the second derivative to zero and solve for \( x \)**:
   \[
   -6x + 84 = 0
   \]
   \[
   6x = 84
   \]
   \[
   x = 14
   \]

4. **Verify the change in concavity**:
   To confirm that \( x = 14 \) is an inflection point, we need to check the concavity on either side of \( x = 14 \).

   - For \( x < 14 \), choose \( x = 13 \):
     \[
     R''(13) = -6(13) + 84 = -78 + 84 = 6 \quad (\text{positive, concave up})
     \]

   - For \( x > 14 \), choose \( x = 15 \):
     \[
     R''(15) = -6(15) + 84 = -90 + 84 = -6 \quad (\text{negative, concave down})
     \]

Since the second derivative changes sign from positive to negative as \( x \) passes through 14, there is indeed an inflection point at \( x = 14 \).

5. **Find the corresponding \( R(x) \)**:
   \[
   R(14) = 10000 - (14)^3 + 42(14)^2 + 600(14)
   \]
   \[
   R(14) = 10000 - 2744 + 42 \cdot 196 + 8400
   \]
   \[
   R(14) = 10000 - 2744 + 8232 + 8400
   \]
   \[
   R(14) = 10000 + 8232 + 8400 - 2744
   \]
   \[
   R(14) = 23888
   \]

Therefore, the inflection point of \( R(x) \) is \( (14, 23888) \).

Would you like more details or have any questions? Here are some other questions you might consider:

1. How do you find the critical points of a function?
2. What is the difference between local maximum and minimum points?
3. How do you determine the concavity of a function?
4. Can you explain the concept of the derivative and its applications?
5. How do you integrate a polynomial function?

**Tip**: When finding inflection points, always check the sign of the second derivative on either side of the candidate point to confirm a change in concavity.

Explore how to find the inflection point for the function R(x) = 10000 - x^3 + 42x^2 + 600x over the interval 0 ≤ x ≤ 20. This detailed solution involves calculating the first and second derivatives, verifying concavity changes, and determining the coordinates of the inflection point.

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