Calculus: Analytic Geometry and Calculus, with Vectors

(^304) Functions, graphs, and numbers
use the result to determine the population of a Utopian country that minimizes
the burdens which individual workers must bear. Hint: Information about
officers in efficient productive organizations is not relevant, but Parkinson laws
may be used.
36 Find the minimum of the function F for which
F(X) =
(OI
[x2 - (x + X)]2 dx. dns.: qI
5.3 Second derivatives, convexity, and flexpoints In Section 3.6
we called attention to the connection between second derivatives and
accelerations. This section shows how second derivatives can be used to
obtain information about functions and their graphs. To begin the pro-
ceedings, we look at Figure 5.31, which shows the graph of a function for
Figure 5.31
which the derivative (or first derivative) f'(x) and the second derivative
(the derivative of the derivative) f "(x) exist when a < x < b. To get
some ideas, we think of the graph as being a road in a vertical plane upon
which we can travel from d to G, and we take the x axis to be at sea level.
During the whole trip, we are always above sea level. The sign of f(x)
gives us this information. At some times during the trip we are going
uphill, and at other times we are going downhill. The sign of f'(x) gives
us this information. As we travel from A to B, from C to D, and from
B to F we are passing over depressions (or pits), andas we travel from
B to C, from D to E, and from F to G weare passing over humps (or peaks).
As we shall see, f"(x) is our source of information about these things and
about points of inflection or flexpoints B, C, D, E, Fat which slopes attain
local extrema.
The two following theorems are obtained by replacing f by f in Theo-
rems 5.26 and 5.27.
Theorem 5.32 If f' is differentiable over a < x < b and a < xo < b,
then f cannot have a flex point at xo unless f"(xo) = 0.
Theorem 5.33 If f is continuous over an interval ao < x S bo and
f"(x) > 0 when ao < x < bo, then f is increasing over the interval ao <
x < bo. If f is continuous over an intervalao < x S bo and f"(x) < 0
when ao < x < bo, then f is decreasingover the interval ao < x < bo.