Chapter 4.2, Problem 107E

(a)

To determine

To show: The functions $f\left(x\right)=\frac{1}{x^{2n}+1}$ are even for n > 0.

(b)

To determine

To show: The graphs of this function $f\left(x\right)=\frac{1}{x^{2n}+1}$ intersect at the points $\left(\pm 1,\frac{1}{2}\right)$ for all positive values of $n$.

(c)

To determine

To show: The inflection points of these functions occur at $x=\pm \sqrt[2n]{\frac{2n-1}{2n+1}}$, for all positive values of $n$.

(d)

To determine

To sketch: The graph of the function $f\left(x\right)=\frac{1}{x^{2n}+1}$ to verify whether the function is even, the graphs of this function $f\left(x\right)=\frac{1}{x^{2n}+1}$ intersect at the points $\left(\pm 1,\frac{1}{2}\right)$ for all positive values of $n$ $x=\pm \sqrt[2n]{\frac{2n-1}{2n+1}}$, for all positive values of $n$.

(e).

To determine

To describe: How the inflection points and the shape of the graphs change as $n$ increases.