Here is the completed table:
x 0123456
F(x) 013431 −0.5EXAMPLE 31
The graph of the function f(t) is shown in Figure N6–14.
FIGURE N6–14
Let Decide whether each statement is true or false; justify your
answers:
(i) If 4 < x < 6, F(x) > 0.
(ii) If 4 < x < 5, F ′(x) > 0.
(iii) F ′′(6) < 0.
SOLUTIONS:
(i) is true. We know that, if a function g is positive on (a, b), then whereas if g is
negative on (a, b), then However, the area above the x-axis between x = 1 and x = 4 is
greater than that below the axis between 4 and 6. Since
it follows that F(x) > 0 if 4 < x < 6.
(ii) is false. Since F ′(x) = f (x) and f (x) < 0 if 4 < x < 5, then F ′(x) < 0.
(iii) is false. Since F ′(x) = f (x), F ′′ (x) = f ′(x). At x = 6, f ′(x) > 0 (because f is increasing).
Therefore, F ′′(6) > 0.