18: The derivative

f ′′′(x) = e^x

is (when taken in absolute value) an increasing function on the relevant range, so its maximum happens at x = 0.5, its value is e^0.5, 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 e^0.5 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

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