Finding Inflection Points (Applied Calc Question)

[Sam]

The problem: Find the inflection points, if any, for the following: f(x) = e^x + x^-1

I know to find inflection points I have to:

1. Compute f''(x)
2. Determine the points in the domain of f for which f''(x) = 0 or f''(x)
does not exist
3. Determine the sign of f''(x) to the left and right of each point x = c
found in step 2. If there is a change in the sign of f''(x) as we move
across the point x = c, then (c, f(c)) is an inflection point of f.

Well, this is what I came up with:

f'(x) = e^x -x^-2
f''(x)= e^x + 2x^-3

Then, I don't know what to do from there because e^x can never be zero, right? but I don't know. My teacher is saying there are inflection points...

Your help is much appreciated!

Sam

[HallsofIvy]

Yes, ex is never 0, but an inflection point is NOT where "ex= 0". It is where f"= ex+ 2/x3= 0.

There is no "algebraic" way to solve that equation but it certainly has solutions: using Newton's method or a hand-dandy graphing calculator, we have a zero of f", and an inflection point, for x approximately 0.926.