The graph of f'(x) is shown for 0=< x =<10. The areas of the regions between the graph of f' and the x-axis are 20, 6, and 4, respectively.

I'm going to describe the graph of f' since I can't post pictures. The first section of the graph goes up (like the sin function) and the area underneath it is 20. Then, it decreases below the x-axis and increases above the x-axis, and the area underneath it is 6. The graph then increases and decreases onto the x-axis, and the area underneath it is 4.

What is the maximum value of f on the closed interval [0,10] of f(0) = 2?
The answer is 22, but I don't know how to get it.

To answer the question you would be looking at the point in the interval 0 < x < 10 which the area's totaled up is greatest.

The point at which the area is greatest is the interval in which the area is 20, if you span it all the way towards x=10, you'd have (20-6+4 = 18) which is clearly lower than 20.

Also the question includes that f(0) = 2, meaning you have an extra +2 to add.

Therefore the greatest maximum value that can be made is the 20 + 2 = 22.

To find the maximum value of f on the closed interval [0, 10], we need to consider the graph of f'.

Let's break down the information given and analyze the graph step by step:

1. The first section of the graph goes up (like the sin function) and the area underneath it is 20.
This means that f' is positive in this region, and the area underneath it indicates the change in the function f in that region. Since the area is positive, f is increasing in this region.

2. Then, the graph decreases below the x-axis and increases above the x-axis, and the area underneath it is 6.
This indicates that f' is negative in this region, and the area underneath represents the change in f in this region. Since the area is negative, f is decreasing in this region.

3. The graph then increases and decreases onto the x-axis, and the area underneath it is 4.
Similar to the previous step, this indicates that f' is positive in this region. The area being positive suggests that f is increasing in this region.

Now, to find the maximum value of f, we need to consider the behavior of f' and determine where f' changes from positive to negative. This is because the maximum value of f occurs when f' changes from increasing to decreasing.

From the information given, we can see that f' changes from positive to negative after the first section where the area is 20. This suggests that the maximum value of f occurs at the end of this interval.

Therefore, the maximum value of f can be found by adding the area underneath the positive section (20) to the initial value of f(0), which is 2.

f(max) = f(0) + area of positive section = 2 + 20 = 22.

Hence, the maximum value of f on the closed interval [0, 10] is 22.

To find the maximum value of f on the closed interval [0,10], you can use the concept of the Fundamental Theorem of Calculus. Here's how you can do it step by step:

1. Recall that the area under a graph represents the integral of the function. In this case, the areas given are 20, 6, and 4, which correspond to the definite integrals of f'(x) on certain intervals.

2. Express the given information as definite integrals using the areas provided. The first area of 20 corresponds to the integral of f'(x) over some interval [a, b]. The second area of 6 corresponds to the integral of f'(x) over another interval [b, c], and the last area of 4 corresponds to the integral of f'(x) over another interval [c, d].

3. Since the graph of f'(x) first goes up and then decreases onto the x-axis, we can conclude that f'(x) is positive for the interval [a, b], negative for the interval [b, c], and positive again for the interval [c, d].

4. Using the given information that f(0) = 2, we can infer that f(x) is an increasing function from x = 0 to x = a (since f'(x) > 0 in this interval). Therefore, the value of f(a) must be less than 2.

5. Similarly, observe that f(x) is a decreasing function from x = a to x = b (since f'(x) < 0 in this interval). Hence, the value of f(b) must be greater than f(a) (which is less than 2).

6. Continuously applying this logic, we find that f(c) must be less than f(b) and f(d) must be greater than f(c) (which is less than f(b)).

7. Combining these observations, f'(x) starts at a positive value, then goes negative, and finally returns to a positive value. This behavior suggests that f(x) is increasing from x = 0 to x = b, decreasing from x = b to x = c, and then increasing again from x = c to x = 10.

8. Since f(x) is increasing on the interval [0, b], we conclude that the maximum value of f(x) within this interval is f(b).

9. Similarly, since f(x) is decreasing on the interval [b, c], we conclude that the maximum value of f(x) within this interval is f(c).

10. Finally, since f(x) is increasing on the interval [c, 10], we conclude that the maximum value of f(x) within this interval is f(10).

11. Therefore, to find the maximum value of f on the closed interval [0,10], we need to compare the values of f(b), f(c), and f(10).

12. However, since the values of b, c, and d are not specified in the given information, we cannot determine exact numerical values for f(b), f(c), or f(10). Nevertheless, we can conclude that the maximum value of f(x) on the interval [0,10] would be the largest among these values.

13. The answer given, 22, is thus plausible if it is determined based on the information given in the question.

In summary, the maximum value of f on the closed interval [0,10] cannot be determined precisely without the exact values of b, c, and d. However, based on the given information, the answer of 22 seems reasonable if derived from f(b), f(c), and f(10).