calculus
posted by Mike .
The graph of f'(x) is shown for 0=< x =<10. The areas of the regions between the graph of f' and the xaxis 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 xaxis and increases above the xaxis, and the area underneath it is 6. The graph then increases and decreases onto the xaxis, 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 (206+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.