Vtyjkyuk

$$beginarrayll textmaximize & displaystyleint_0^1 x , f(x) , mathrm dx\ textsubject to & displaystyleint_0^1 f(x) , mathrm dx = 1\ & displaystyleint_0^1 x^2 f(x) , mathrm dx = 1\ & f(x) geq 0 quad forall x in [0,1]endarray$$

A few years ago I studied calculus of variations but for some reason I keep chasing my tail on this problem. If my recollection is even close to the mark we start with
$$L=int_0^1 left( f(x) cdot x right) dx + lambda_1 cdot left( int_0^1 left( f(x) right) dx - 1right) + lambda_2 cdot left( int_0^1 left( f(x) cdot x^2 right) dx - 1right)$$

And then Euler Lagrange drops $f$ completely and gives

$$x+lambda_1 +lambda_2 x^2=0$$

If the slack constraints are functional

$$L=int_0^1 left( f(x) cdot x right) + lambda_1(x) cdot left( left( f(x) right) - 1right) + lambda_2(x) cdot left( left( f(x) cdot x^2 right) - 1right) dx$$

gives

$$x+lambda_1(x) +lambda_2(x) x^2=0$$

But I'm not sure if from here how we enforce the $lambda$ partials, which if they're just taken directly seem to contradict each other...

Conceptually there should be a solution. I'd appreciate some tips on this refresher.

Edit:

Only solution to constraints is discontinuous. Poorly posed. What about... The version where the expected value of X is to be minimized such that X>0 and the second Central moment (variance) is 1? The PDF of that I think hits the same roadblocks I hit above but is a nontrivial computation. Goal to find the PDF $f(x)$ on $x>0$.

The problem is impossible as stated with $f geq 0$. The constraints $int_0^1 f(x), dx = int_0^1 x^2 f(x), dx = 1$ may be combined as $int_0^1 (1-x^2) f(x) = 0$. As $1-x^2$ is nonnegative on $[0, 1]$ and positive except at $1$, this constraint is impossible except if $f$ is the Dirac distribution $delta(x-1)$.

If the restriction $f geq 0$ is relaxed, there is no maximum. Let $F$ be an antiderivative of $f$ such that $F(0) = 0$, $F(1) = 1$. In this case, integrating by parts gives the maximized quantity as beginalign* int_0^1 x f(x), dx &= left. x F(x)right|_0^1 - int_0^1 F(x), dx \ &= 1 - int_0^1 F(x), dxendalign* so we have to minimize $int_0^1 F(x), dx$ with the constraints $F(0) = 0$, $F(1) = 1$, and beginalign* 1 &= int_0^1 x^2 f(x), dx \ &= left.x^2 F(x)right|_0^1 - 2 int_0^1 x F(x), dx \
&= 1 - 2 int_0^1 x F(x), dx \
0 &= int_0^1 x F(x), dx endalign*
but we can always decrease $int_0^1 F(x)$ and preserve $int_0^1 x F(x)$ by decreasing the values of $F$ near $x=0$ and increasing them a lesser amount near $x = 1$ (construction of an explicit example is left as an exercise to the reader).