How can I find the number of points of inflection for a function whose second derivative is .5 + cosx - e^-x on the interval x is greater than or equal to 0 and less than or equal to 20?

Answer 1

Find the points in that interval where the second derivative is zero. Instead of solving for those points, I suggest plotting a curve of f''(x) and counting the number of zero crossings between 0 and 20.

There may be about 4 such points, at x values near 1.9, 4.1, 8.5 and 14.7. Try it yourself; don't trust my numbers.