What is the maximum of $f(x)=1/(\lambda^k(x+\lambda)^2)$

Question

What is the maximum of

$$\begin{aligned} f:(0,\infty) &\to\mathbb{R} \\ x&\mapsto\frac{1}{\lambda^k(x+\lambda)^2} \end{aligned}$$ where $\lambda>0$ and $k\ge 1$?

Am I naive to think that it is simply $1/\lambda^{k+2}$ without performing any differentiation tests?

In any case, solving $$f'(x)=-\frac{2}{\lambda^k(x+\lambda)^{3}}=0$$ for $x$ seems very nontrivial.

Answer 1

The function does not attain its maximum on $(0,\infty)$. However, its supremum is as you say $\lambda^{-(k+2)}$. It is attained at $x=0$, but $0\notin(0,\infty)$.