Is the circle the ellipse with minimal perimeter? Consider a family of ellipses parametrized as $(a \cos t, a^{-1} \sin t)$, $a\in (0,\infty)$ and $t\in(0,2\pi]$, such that them all have the same area, $\pi$.
I wish to prove that (only) the circle has the minimal perimeter. So considering the function
\begin{align}
P(a)=\int _0^{2\pi}\sqrt{a^2\sin^2 t+a^{-2}\cos^2t}\:dt
\end{align}
how can one prove that
\begin{align}
P'(a)=\int _0^{2\pi}\frac{a\sin^2 t-a^{-3}\cos^2t}{\sqrt{a^2\sin^2 t+a^{-2}\cos^2t}}\:dt
\end{align}
vanishes only for $a=1$ ?
 A: Rewrite $P'(a)$ as ${2\over a^2}f(a)$, where:
$$
f(a)=\int_0^\pi{a^4\sin^2t-\cos^2t\over
\sqrt{a^4\sin^2t+\cos^2t}}dt.
$$
We then have:
$$
f'(a)=\int_0^\pi{\frac{2 a^3 \sin ^2(t) \left(a^4 \sin ^2(t)+
3 \cos ^2(t)\right)}{\left(a^4 \sin^2(t)
+\cos ^2(t)\right)^{3/2}}}dt>0.
$$
Hence $f(a)$ is increasing and vanishes only for $a=1$.
A: $\newcommand{\bbx}[1]{\,\bbox[15px,border:1px groove navy]{\displaystyle{#1}}\,}
 \newcommand{\braces}[1]{\left\lbrace\,{#1}\,\right\rbrace}
 \newcommand{\bracks}[1]{\left\lbrack\,{#1}\,\right\rbrack}
 \newcommand{\dd}{\mathrm{d}}
 \newcommand{\ds}[1]{{\displaystyle #1}}
 \newcommand{\expo}[1]{\,\mathrm{e}^{#1}\,}
 \newcommand{\ic}{\mathrm{i}}
 \newcommand{\on}[1]{\operatorname{#1}}
 \newcommand{\pars}[1]{\left(\,{#1}\,\right)}
 \newcommand{\partiald}[3][]{\frac{\partial^{#1} #2}{\partial #3^{#1}}}
 \newcommand{\root}[2][]{\,\sqrt[#1]{\,{#2}\,}\,}
 \newcommand{\sr}[2]{\,\,\,\stackrel{{#1}}{{#2}}\,\,\,}
 \newcommand{\totald}[3][]{\frac{\mathrm{d}^{#1} #2}{\mathrm{d} #3^{#1}}}
 \newcommand{\verts}[1]{\left\vert\,{#1}\,\right\vert}$
Hereafter, $\ds{\on{E}\ \mbox{and}\ \on{K}}$ are the Complete Elliptic Integral and the Complete Elliptic Integral of the First Kind, respectively.
\begin{align}
& \left\{\begin{array}{rcl}
\ds{\on{P}\pars{a}} & \ds{=} &
\ds{{4\on{E}\pars{1 - a^{4}} \over a^{2}}}
\\[2mm]
\ds{\on{P}'\pars{a}} & \ds{=} &
\ds{{4 \over a^{2}\pars{a^{4} - 1}}\ \times}
\\ &&
\ds{\bracks{\pars{1 + a^{4}}\on{E}\pars{1 - a^{4}} -2a^{4}\on{K}\pars{1 - a^{4}}}}
\end{array}\right.
\end{align}
$\ds{\on{P}\pars{a}\ \mbox{and}\
\on{P}'\pars{a}}$ pictures, respectively:


