To analyze this problem, we are tasked with identifying the local maxima and local minima of the function $g(x) = \int_0^x f(t) \, dt$, where the graph of $f(x)$ is provided.

Key Properties:

1. $g'(x) = f(x)$, meaning the slope of $g(x)$ is determined by $f(x)$.
2. Local maxima/minima of $g(x)$ occur where $f(x) = 0$ (critical points) and where $f(x)$ changes sign:
   • $g(x)$ has a local maximum where $f(x)$ changes from positive to negative.
   • $g(x)$ has a local minimum where $f(x)$ changes from negative to positive.

Observing the Graph:

• The graph shows $f(x)$ crossing the x-axis at several points in the interval $[0, 10]$.
• We now identify these critical points and analyze the sign changes of $f(x)$:

1. $f(x) = 0$ at $x \approx 1.5$, $x \approx 4.5$, $x \approx 7$, and $x \approx 9.5$.

Sign Analysis and Classification:

• Between $x = 0$ and $x \approx 1.5$, $f(x) > 0$: $g(x)$ increases.
• At $x \approx 1.5$, $f(x)$ changes from positive to negative: local maximum.
• Between $x \approx 1.5$ and $x \approx 4.5$, $f(x) < 0$: $g(x)$ decreases.
• At $x \approx 4.5$, $f(x)$ changes from negative to positive: local minimum.
• Between $x \approx 4.5$ and $x \approx 7$, $f(x) > 0$: $g(x)$ increases.
• At $x \approx 7$, $f(x)$ changes from positive to negative: local maximum.
• Between $x \approx 7$ and $x \approx 9.5$, $f(x) < 0$: $g(x)$ decreases.
• At $x \approx 9.5$, $f(x)$ changes from negative to positive: local minimum.

Final Answer:

• Local maxima: $x \approx 1.5$, $x \approx 7$
• Local minima: $x \approx 4.5$, $x \approx 9.5$

Would you like a more detailed explanation or help with related concepts?

5 Follow-Up Questions:

1. Can you explain why $g(x)$ is an antiderivative of $f(x)$?
2. How would the integral $g(x)$ behave if $f(x)$ stayed entirely positive on $[0, 10]$?
3. What would happen to the maxima and minima if $f(x)$ were shifted upward or downward?
4. How does the Fundamental Theorem of Calculus relate to this problem?
5. Can you determine the total change in $g(x)$ over the interval $[0, 10]$?

Tip:

Always link the critical points of a graph to where its derivative (or related function) changes sign—this is essential for finding maxima and minima!