Let's analyze the given graph and solve each part of the problem.

Problem Statement

1. a) Determine $f(1)$ using the graph of $f$.

   • To find $f(1)$, locate $x = 1$ on the graph and read the corresponding $y$-value.
   • Observing the graph, when $x = 1$, the $y$-value appears to be approximately 1.
   • Answer: $f(1) = 1$.

2. b) Solve the equation $f(x) = 0$ using the graph (note: remember all solutions).

   • To solve $f(x) = 0$, find the points where the graph intersects the $x$-axis (i.e., where $y = 0$).
   • From the graph, the function intersects the $x$-axis at approximately $x = -3$, $x = 0$, and $x = 2$.
   • Answer: The solutions to $f(x) = 0$ are $x = -3$, $x = 0$, and $x = 2$.

3. c) Determine the local extrema of $f$.

   • Local extrema occur at the peaks (local maxima) or valleys (local minima) of the graph.
   • Observing the graph:
     • There is a local maximum at approximately $x = -2$, with $f(x) \approx 4$.
     • There is a local minimum at approximately $x = 0$, with $f(x) = 0$.
   • Answer: Local maximum at $x = -2$ with $f(-2) = 4$ and local minimum at $x = 0$ with $f(0) = 0$.

Would you like further explanations or details on any specific part of the solution?

Additional Related Questions:

1. How can we find the derivative of $f(x)$ to determine extrema analytically?
2. What does the slope of $f(x)$ indicate about increasing or decreasing intervals?
3. How do we classify extrema (local maximum vs. minimum) using the first or second derivative tests?
4. How can we verify the exact values of intersections and extrema using calculus?
5. Can we determine the concavity of $f(x)$ from the graph alone?

Tip:

For functions with multiple turning points, extrema can be identified visually on a graph, but derivatives are often used for precise calculation.