The graph in the image represents the derivative $f'$ of a function $f$. You're asked to find:

1. The $x$-value at the peak (or extremum) of the function $f$.
2. The second zero of $f$, with the information that one zero occurs at $x = 1$.

Analyzing the Graph of $f'$:

• The graph shows a straight line with a negative slope, crossing the x-axis at two points.
• These points represent where the slope of $f$ (i.e., $f'$) is zero, indicating potential maxima or minima of $f$.

Task 1: Finding the $x$-value at the peak of $f$

• The peak of $f$ occurs where $f'(x) = 0$. According to the graph, $f'(x) = 0$ at $x = 1$.
• Since this is given, $x = 1$ is the location of the peak of $f$.

Task 2: Finding the second zero of $f$

• The second zero of $f'$ corresponds to another place where $f'(x) = 0$, potentially indicating another critical point (like a minimum) for $f$.
• Looking at the graph, the line crosses the x-axis again at $x = 5$, so the second zero of $f$ occurs at $x = 5$.

Thus:

• The $x$-value of the peak of $f$ is $x = 1$.
• The second zero of $f$ is $x = 5$.

Would you like more details on these calculations?

Related questions:

1. How can you use the second derivative test to determine if $x = 1$ is a maximum or minimum?
2. What would happen if the slope of $f'$ was positive instead of negative?
3. What can you deduce about the concavity of $f$ based on $f'$?
4. How does knowing the derivative graph help in sketching the original function $f$?
5. How would the graph change if there was an inflection point for $f$?

Tip:

Remember, the zeros of the derivative indicate critical points where the original function might have maxima, minima, or inflection points.