How do I find the value of $p$

Question

Consider the function $h(x) = (x + p) \arctan x$, where $p$ is a real constant and affects the asymptotic behaviour of $h(x)$ as well as its local behaviour. What values of $p$ can give $h(x)$ inflection points? Can somebody teach me where to start off?

Answer 1

Since $\frac{\mathrm {d}}{\mathrm {d} x} \arctan x = \frac{1}{x^2+1}$, then we have by the product rule and quotient rule:

$$h(x) = (x + p) \arctan x$$ $$h'(x) = \arctan x + \frac{x+p}{x^2+1}$$ $$h''(x) = \frac{1}{x^2+1} + \frac{(x^2+1) - (x+p)(2x)}{(x^2+1)^2}$$ $$=\frac{2(x^2+1) -2x^2-2xp}{(x^2+1)^2} = \frac{2-2xp}{(x^2+1)^2}$$

Since $(x^2+1)^2 = x^4+2x^2+1 ≥ 0+0+1$, $(x^2+1)^2 \ne 0$ for all real $x$.

Therefore, we have $2-2xp = 0 \implies xp=1, x = \frac{1}{p}$. This means that there will always be an inflection point at $x = \frac{1}{p}$ unless $\frac{1}{p}$ is undefined, which is $p=0$.