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The problem involves analyzing the function \( g(x) = \int_0^x f(t) \, dt \), where \( f(t) \) is the given graph. Let's approach each question step-by-step:

### (a) Local Maxima and Minima of \( g(x) \)
To find the local maxima and minima of \( g(x) \), we need to understand where the derivative \( g'(x) = f(x) \) changes sign (i.e., where \( f(x) \) crosses the \( x \)-axis or peaks).

1. **Local Minima ( \( x_{\text{min}} \) ):**
   - Local minima of \( g(x) \) occur where \( f(x) \) changes from negative to positive (crosses the \( x \)-axis from below).
   - Observing the graph of \( f(t) \):
     - The smaller \( x \)-value for a local minimum seems to be around \( t = 10 \).
     - The larger \( x \)-value for a local minimum is around \( t = 30 \).

   Thus,
   \[
   x_{\text{min}} = 10 \quad \text{and} \quad x_{\text{min}} = 30
   \]

2. **Local Maxima ( \( x_{\text{max}} \) ):**
   - Local maxima of \( g(x) \) occur where \( f(x) \) changes from positive to negative.
   - Observing the graph of \( f(t) \):
     - The smaller \( x \)-value for a local maximum appears around \( t = 5 \).
     - The larger \( x \)-value for a local maximum appears around \( t = 35 \).

   Thus,
   \[
   x_{\text{max}} = 5 \quad \text{and} \quad x_{\text{max}} = 35
   \]

### (b) Absolute Maximum of \( g(x) \)
The absolute maximum of \( g(x) \) occurs at the point where the cumulative area under \( f(t) \) from \( t = 0 \) to \( x \) reaches its highest positive value.

Looking at the graph of \( f(t) \), the area under the curve seems to accumulate the most by around \( t = 5 \). Therefore:
\[
x = 5
\]

### (c) Concave Downward Intervals for \( g(x) \)
The concavity of \( g(x) \) is determined by the sign of \( f'(x) \). \( g(x) \) is concave downward where \( f(x) \) is decreasing.

From the graph, \( f(t) \) is decreasing in the interval approximately from \( t = 5 \) to \( t = 15 \) and again from \( t = 30 \) to \( t = 40 \).

Thus, the interval where \( g(x) \) is concave downward is:
\[
(5, 15) \cup (30, 40)
\]

### (d) Sketching the Graph of \( g(x) \)
For sketching \( g(x) \), use the following observations:
- \( g(x) \) starts at 0 when \( x = 0 \).
- \( g(x) \) increases where \( f(x) \) is positive and decreases where \( f(x) \) is negative.
- Local maxima and minima of \( g(x) \) correspond to where \( f(x) \) crosses the \( x \)-axis or reaches peaks and troughs.
- Concave up and concave down behavior corresponds to intervals where \( f(x) \) is increasing or decreasing, respectively.

Would you like more detailed steps on any part of this solution or assistance with the graph of \( g(x) \)?

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### Follow-Up Questions:
1. How can we determine concavity from the second derivative in general?
2. Why does the integral of a function \( f(t) \) provide insight into the accumulation of areas under \( f(t) \)?
3. How does the Fundamental Theorem of Calculus relate the function \( g(x) \) and its derivative?
4. Why does \( g(x) \) reach a local maximum where \( f(x) \) crosses from positive to negative?
5. How would we find the exact value of \( g(x) \) at specific points if \( f(t) \) were given as a formula?

### Tip:
When analyzing the integral function \( g(x) = \int_0^x f(t) \, dt \), keep in mind that \( g(x) \) reflects the *net area* under \( f(t) \) from 0 to \( x \). Positive areas increase \( g(x) \), and negative areas decrease it.

Let g(x) = ∫[0 to x] f(t) dt, where f is the function whose graph is shown. (a) At what values of x do the local maximum and minimum values of g occur? (b) Where does g attain its absolute maximum value? (c) On what interval is g concave downward? (d) Sketch the graph of g.

Explore how to analyze the function g(x) = ∫[0 to x] f(t) dt by examining the local extrema, absolute maximum, and concavity intervals of g(x). This problem covers integral calculus concepts, including the Fundamental Theorem of Calculus, concavity, and graph analysis, with a detailed step-by-step solution. Ideal for students in grades 11-12 or introductory calculus.

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