The graph of f ′ (x), the derivative of f(x), is continuous for all x and consists of five line segments as shown below. given f (5) = 10, find the absolute minimum value of f (x) over the interval [0, 5].
a.3
b.0
c.7
d.17

According to the graph, we have for

is positive for all , which means is strictly increasing on the interval [0, 5], and so its minimum value on this interval occurs at .

We can integrate along each subinterval above to find :

is continuous, which means is also continuous. This means

and

Given that , we have

so that over we have . We know the minimum value occurs at , so it is 3 and the answer is A.