1, the sign of f ′ changes from positive to negative; as x passes through 3, the sign of f ′ changes
from negative to positive. Therefore f (1) is a local maximum and f (3) a local minimum. Since f
changes from concave down to concave up at x = 2, there is an inflection point on the graph of f
there.
These conclusions enable us to get the general shape of the curve, as displayed in Figure N6–
11a.
FIGURE N6–11aFIGURE N6–11b
All that remains is to evaluate f (x) at x = 1, 2, and 3. We use the Fundamental Theorem of
Calculus to accomplish this, finding f also at x = 4 and 5 for completeness.
We are given that f (0) = 1. Then
where the integral yields the area of the triangle with height 2 and base 1;
where the integral gives the area of a quadrant of a circle of radius 1 (this integral is negative!);
where the integral is the area of the triangle with height 1 and base 1;