Question about finding the maximum value of an integral plotted on a graph

Here's a screenshot of the graph and the question:

http://oi55.tinypic.com/25jduh4.jpg

The possible answer choices are:
A) 4
B) 6
C) 7
D) 8
E) some number other than 0, 4, 6, 7, 8

I chose my answer as 7 because the area under the graph represents the integral g(x), and the integral starts at 7, so at x = 7, the value of the integral would be 0, no? The answer key, however, says it's 6. Can someone explain?

Thanks in advance!

You know that when the derivative of a function crosses the x-axis (i.e. is at (x,0), where x is any value on the x-axis) that function is at a maximum or a minimum. If we draw the function y = -x^2, you can see were the bell shape at the top; if you were to take the gradient of the tangent to the graph of y at the maximum point of the graph, you would see that the gradient, or derivative of that line is 0 - this serves as proof for the statement made above.
For the function f(t), where integral(f(t)dt) = g(x), f(t) crosses the x-axis (or t-axis in this case) at t = 6.
In addition, a rule for figuring out whether the point on the function is a maximum or minimum, find the second derivative (i.e. the derivative of the derivative) at that point. If the value of the second derivative is negative, you know that the function is at a maximum, whereas if the value of the second derivative is at a positive, the function is at a maximum. The reason for this is that when the second derivative is negative, you know that the gradient of the derivative function is also negative. If the derivative function is crossing the x-axis as it's gradient is negative, this means that the actual function's gradient is going from positive to negative. This shows that it is a maximum point, and not a minimum.

I hope this was of use to you. Normally, I would agree with your assumption, since, according to the question, g(x) is only true for values of t greater than or equal to 0. However, if you forget the given information and assume that g(x) is true for all values of t and x, then the method I have described above should, hopefully, help you.

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Well, you found the minimum value. What's the maximum?

The graphic displays a plot of f(x) which is the derivative of g(x).

The minimum and maximum values of the function, if there are any, will occur where the derivative is zero or at the endpoints of the domain.

Look at the graph. The derivative is zero at x = 0 and x= 6.

Test those points (as well as the endpoints) to see which is a minimum and which is a maximum.

You could take the second derivative but I suspect you can figure it out without going to that much trouble.