18: The derivative
f ′′′(x) = ex
is (when taken in absolute value) an increasing function on the relevant
range, so its maximum
happens at x = 0.5, its value is
e0.5,
that is, the root of e.
This is unpleasant, since we do not know how much it is. We estimated it to
be a bit more than 1.6 short while ago, but we do not know how good this
estimate is (we are now trying to figure out is reliability).
However, note we do not actually need to know this value in Lagrange's
estimate, we would be happy with some upper estimate. Since 22 is
more than e, the root of e is definitely less than 2. Could we
make a better estimate? For instance,
1.82 = 3.24 > e, so
the root of e is surely less than 1.8. Could we go still lower?
Probably yes, but if our estimate of e0.5 is any good, then
not by much, so we leave it. We will use 1.8 as the upper estimate, actually
9/5 will be easier to work with. Thus
![](gif4/ecc4cb4b.gif)
Answer