A monster problem for Halloween.

A while ago I suggested you look at , which using the dominance idea is zero. Of course your students may try graphing or a table. Here’s the graph done by a TI-Nspire CAS. Note the scales.

This is not the way to go. Since the function is increasing near the origin, but the limit at infinity is zero there must be a maximum point where the function starts decreasing. And as the expression can never be negative once *x* > 1, there must be a point of inflection where the graph becomes concave up and can thereafter approach the *x*-axis from above as a horizontal asymptote. The maximum can be found by hand which makes for some great algebra manipulation practice:

Setting this equal to zero and solving gives

The second derivative is

and is zero when *x* =

Okay, I skipped a few steps here, but you can challenge your students with that. Since we’re really interested in the solution here more than the solving ,this is really a good place to use a CAS calculator.

The first line in the figure above defines the function to save typing it each time. The second line finds the *x-*coordinate of the maximum point (how do we know this is a maximum?) and the third finds the *x*-coordinate of the point of inflection. Much simpler this way!

Take a minute to consider the numbers. **They are BIG!** In fact, if the units on our graph paper are centimeters then the maximum point is a little over **5,480 light-years** away from the origin! The point of inflection is about 2.665 times farther at more than **14,607 light-years **away!

Meanwhile the maximum value is only 91.9699 cm. That’s right *centimeters*, less than a meter. And the *y*-coordinate of the point of inflection is about 91.9524 cm. A drop of 0.0175 cm. in a horizontal distance of a little over 9,127 light-years.

Some problems are a lot less scary if done with technology.

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