_{0}

^{x }g(t)dt on the interval [0, 5], where g(t) is given by the graph below. (Remark: The part of the graph over [3, 5] is a quarter circle, and the part of the graph over [0, 3] is a line going straight up and to the right.)

4. Find the absolute maximum and minimum value of f(x)=S_{0}^{x }g(t)dt on the interval [0, 5], where g(t) is given by the graph below. (Remark: The part of the graph over [3, 5] is a quarter circle, and the part of the graph over [0, 3] is a line going straight up and to the right.)

Note: I'm using "S" as the definite integral sign.

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Your function is piecewise continuous!

The first piece is just a straight line, y=x-1, for 0≤x<3.

If the second piece really is a quarter circle, you will need its equation. The equation for a circle centered at (3,0) of radius 2 is (x-3)² + y² = 4, so the quarter circle as a function is

y = sqrt(4-(x-3)²) = sqrt(-x²+6x-5), for 3≤x≤5.

So the piecewise function g(t) is

g(t) = { (t-1) 0≤t<3

{ sqrt(-t²+6t-5) 3≤t≤5

Now f(x)=∫_{0}^{x} g(t) dt, which will also be piecewise continuous. It's the area between the graph of g(t) and the t-axis, counting areas below the t-axis as negative. To find the absolute extrema of f(x), you do not need to carry out the integral, assuming you know the area of a circle and of a triangle. For which x is the area smallest? For which x is the area largest? What are the min and max values of these areas?

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