To determine the open intervals on which the function $f(x) = x^2 - 8x$ is increasing, decreasing, or constant, we need to analyze the function using calculus principles, specifically by finding the first derivative and determining critical points.

Step 1: Find the derivative of $f(x)$

The derivative of the function $f(x) = x^2 - 8x$ is: $f'(x) = 2x - 8$

Step 2: Find the critical points

To find the critical points, we set $f'(x) = 0$: $2x - 8 = 0$ $2x = 8 \quad \Rightarrow \quad x = 4$ Thus, the critical point is $x = 4$.

Step 3: Analyze the intervals

The critical point $x = 4$ divides the x-axis into two intervals: $(-\infty, 4)$ and $(4, \infty)$. We need to determine where the function is increasing or decreasing by testing points in these intervals.

• For $x < 4$ (e.g., $x = 0$): $f'(0) = 2(0) - 8 = -8$ Since $f'(0) < 0$, the function is decreasing on the interval $(-\infty, 4)$.

• For $x > 4$ (e.g., $x = 5$): $f'(5) = 2(5) - 8 = 2$ Since $f'(5) > 0$, the function is increasing on the interval $(4, \infty)$.

Step 4: Conclusion

• The function is decreasing on the interval $(-\infty, 4)$.
• The function is increasing on the interval $(4, \infty)$.
• There is no interval where the function is constant (which is why "DNE" is the correct answer for "constant").

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Related questions:

1. How do you determine the concavity of a function?
2. What does the second derivative test tell you about a function's graph?
3. How can you find the maximum or minimum values of a quadratic function?
4. What are the steps to sketch the graph of a parabola?
5. How do critical points help in determining a function’s behavior?

Tip:

Always double-check the intervals where the first derivative changes sign to accurately identify increasing and decreasing behavior!