- Define the following functions:
for
. Prove that for
and
the inequalities
hold. Proceed by examining the signs of the derivatives
and
. When
these are equalities.
- Draw the graphs of the functions
Interpret the inequalities in part (a) geometrically.
- First, we compute the derivatives of the functions,
Then, considering the derivative of
,
Therefore, the function
has a minimum at
. Since we can directly evaluate
, this means
for
and
. Therefore,
.
Next, looking at the derivative of
,
Therefore, the function
has a minimum at
. Since
this implies
for
and
. Thus,
Putting these two pieces together we have established the requested inequalities:
-
The graph of the two functions is:
The inequalities in part (a) imply that the graph of
must lie strictly between the graphs of
and
shown above (and so
).