Given the function $f(x) = 2x^5 - 604x - 725x^3 - 2875x^2$, we need to address the following points:

Step 1: Rewrite and Organize the Function

To simplify analysis, write $f(x)$ in descending order of powers:

$f(x) = 2x^5 - 725x^3 - 2875x^2 - 604x$

Part (a): Find Values of $x$ Where $f(x)$ Has a Relative Maximum or Minimum

To locate relative maxima and minima, we find the critical points by calculating the derivative and setting it to zero.

1. Compute the First Derivative $f'(x)$:

   $f'(x) = 10x^4 - 2175x^2 - 5750x - 604$

2. Set $f'(x) = 0$:

   Solve $10x^4 - 2175x^2 - 5750x - 604 = 0$ to find the critical points. This is a polynomial equation that can be solved numerically, as it may not have simple analytical solutions.

3. Analyze the Critical Points: Evaluate the second derivative at these points to determine if they correspond to maxima or minima.

Part (b): Find Values of $x$ on $[0, 30]$ Where $f(x)$ Reaches Absolute Max or Min

1. Evaluate $f(x)$ at Critical Points in $[0, 30]$ Found in Part (a).

2. Evaluate $f(x)$ at the Endpoints $x = 0$ and $x = 30$:

   • Find $f(0)$.
   • Find $f(30)$.

3. Compare Values: The absolute maximum and minimum on $[0, 30]$ will be among the values found at the endpoints and the critical points in the interval.

Part (c): Find Points of Inflection

Points of inflection occur where the concavity of the function changes, which happens when the second derivative $f''(x)$ changes sign.

1. Compute the Second Derivative $f''(x)$:

   $f''(x) = 40x^3 - 4350x - 5750$

2. Set $f''(x) = 0$:

   Solve $40x^3 - 4350x - 5750 = 0$ to find potential points of inflection.

3. Analyze for Sign Changes: Check intervals around these solutions to see if $f''(x)$ changes sign, indicating a change in concavity.

Summary of Results

1. Relative Max/Min: Solve $f'(x) = 0$.
2. Absolute Max/Min on $[0, 30]$: Evaluate $f(x)$ at endpoints and critical points in the interval.
3. Inflection Points: Solve $f''(x) = 0$ and test for sign changes in $f''(x)$.

Would you like the detailed solutions for each step or assistance with the calculations?

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Follow-Up Questions

1. How do you determine if a critical point is a maximum or minimum?
2. What methods are available for solving polynomial equations?
3. What is the difference between relative and absolute extrema?
4. How can the second derivative test confirm points of inflection?
5. Why do we check endpoints when finding absolute extrema on an interval?

Tip: In optimization problems, critical points in an interval are essential for identifying relative extrema.