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**Theorem 346**

Suppose *f* is continuous on an interval *I* and differentiable at
every interior point of *I*.

- 1.
- If
*f*'(*x*) > 0 on the interior of*I*, then*f*is increasing on*I*. - 2.
- If
*f*'(*x*) < 0 on the interior of*I*, then*f*is decreasing on*I*.

This theorem says that we can determine when a function is
increasing or decreasing by solving the inequalities *f*'(*x*) > 0 and
*f*'(*x*) , 0. In practice, we usually work with functions having
continuous derivatives, which means that *f*' can change sign only at
a point where *f*'(*x*)=0. For example, consider *f*(*x*) = *x ^{2}*. The
derivative is

- 1.
- Find the critical points of
*f*. Note that according to the definition in the text, critical points of*f*are points where either*f*' is zero, the derivative doesn't exist, or endpoints of*I*if*f*is defined on a finite interval*I*. - 2.
- The critical points divide the domain of
*f*into subintervals on which the sign of*f*' is constant. Check the sign of*f*' at one interior point on each subinterval. If it is positive,*f*is increasing on that subinterval. If it is negative,*f*is decreasing on that subinterval.

The second derivative, *f*''(*x*) also provides
information about the shape of the curve in terms of what is called
concavity. Concavity can also be defined in several ways. Geometrically,
it can be said that the graph of *f* is concave up near a point
*x*=*a* if the tangent line at *x*=*a* lies below the graph of *f* on some
open interval containing *a* and is concave up if the tangent line
lies above the graph of *f* on some open interval containing
*c*. Algebraically, concavity is most often defined by saying that *f*
is concave up on an interval *I* if *f*' is increasing on *I* and is
concave down on *I* if *f*' is decreasing on *I*. Using the theorem
above and remembering that *f*'' is the derivative of *f*' gives the
following result.

**Theorem 403**

Suppose *f* be twice differentiable on the open interval *I*.

- 1.
- If
*f*''(*x*) > 0 on*I*, then*f*is concave up on*I*. - 2.
- If
*f*''(*x*) < 0 on*I*, then*f*is concave down on*I*.

This means that we can find where *f* is concave up and concave down
using the same procedure on *f*'' that we used on
*f*'. That is, we first find all of values of *x* for which *f*''(*x*)=0, or
*f*''(*x*) doesn't exist. Including the endpoints, if our domain is a
finite interval, these values of *x* are the endpoints of distinct
subintervals on which the sign of *f*'' is constant. Checking the sign
of *f*'' at one point in the interior of each subinterval determines
the concavity of *f* on that subinterval.

- 1.
- Suppose
*f*id continuously differentiable and*c*is a number such that*f*'(*c*)=0. Is it necessarily true that*f*is increasing in some interval on one side of*c*and decreasing in some interval on the other side of*c*? - 2.
- For the following functions, find the intervals on which they
are increasing and the intervals on which they are decreasing.
- (a)
*f*(*x*) =*x*-4^{3}*x*+1.- (b)
*f*(*x*) =*x*-23^{4}*x*+59^{3}*x*+23^{2}*x*-60.- (c)
- .
- (d)
*f*(*x*) =*x*+*sin*(*x*).- (e)

- 3.
- For the following functions, find the intervals on which they
are concave
upward and the intervals on which they are concave downward. Make sure
that you
explain how you got your results.
- (a)
*f*(*x*) =*x*-4^{3}*x*+1.- (b)
*f*(*x*) =*x*-23^{4}*x*+59^{3}*x*+23^{2}*x*-60.- (c)
- .
- (d)
*f*(*x*) =*x*+*sin*(*x*).- (e)

2/15/2000