To address this problem, let's first analyze the function and graph it:

Function:

$f(x) = 5x$, for $x \geq 5$

This is a linear function with a constant slope of 5, which means the function increases as $x$ increases. Since the function is linear, there will be no local maximum or minimum points (those are typically present in functions with curves, like quadratic or trigonometric functions). However, we can identify the absolute maximum or minimum based on the domain restriction, $x \geq 5$.

Steps:

1. Graph of $f(x) = 5x$ for $x \geq 5$:

   • The function is linear and starts at $x = 5$. At $x = 5$, the function value is $f(5) = 5(5) = 25$.
   • As $x$ increases beyond 5, the function keeps increasing linearly.

2. Absolute maximum: Since the function increases without bound as $x \to \infty$, there is no absolute maximum. The function does not peak; it keeps rising indefinitely.

3. Absolute minimum: The function is restricted to $x \geq 5$. The absolute minimum occurs at the left endpoint, $x = 5$, where $f(5) = 25$.

4. Local maximum and minimum values: For linear functions like this, there are no turning points, so there are no local maxima or minima within the domain.

Conclusion:

• Absolute maximum value: DNE (the function increases without bound)
• Absolute minimum value: 25
• Local maximum value(s): DNE (linear functions do not have local maximums)
• Local minimum value(s): DNE (linear functions do not have local minimums)

Would you like me to go into more detail about any step?

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Here are 5 related questions for further understanding:

1. How would the graph change if the slope of the line were negative?
2. How can we find local extrema for non-linear functions?
3. What happens to the absolute minimum and maximum if the domain is changed?
4. How would you determine the extrema if the function included a quadratic term?
5. Can a linear function ever have a local maximum or minimum?

Tip: When dealing with linear functions, extrema (maximum or minimum points) only occur at the boundaries of the domain.