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To find the point of diminishing returns for the given revenue function:

\[
R = -1.71 \left(x^3 - 12x^2 - 6\right)
\]

we need to locate the inflection point of this function, as the hint suggests. Inflection points occur where the second derivative changes sign, indicating a change in the concavity of the graph, which can signify the point where revenue growth starts to slow down (diminishing returns).

### Step-by-Step Solution

1. **First Derivative \( R'(x) \):**
   Find the first derivative \( R'(x) \) to understand the rate of change of revenue with respect to \( x \) (advertising).

   \[
   R(x) = -1.71 \left(x^3 - 12x^2 - 6\right)
   \]
   Using the chain rule:
   \[
   R'(x) = -1.71 \cdot (3x^2 - 24x)
   \]
   Simplifying:
   \[
   R'(x) = -1.71 \cdot 3x^2 + 1.71 \cdot 24x
   \]


Find the point of diminishing returns for the given revenue function: R = -1.71(x^3 - 12x^2 - 6), where R stands for revenue and x is the amount spent on advertising (in thousands of dollars).

Learn how to find the point of diminishing returns for a revenue function R = -1.71(x^3 - 12x^2 - 6). This calculus problem involves identifying the inflection point to determine when revenue growth starts to slow. Suitable for high school students in Grades 11-12.

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