Dear Toni, the formula after the phrase "rearranging the terms and taking" should be corrected since in second, third and fourth terms the multiplier $x$ is missing.

Thanks! The problem is solved.

Thanks! For me it is also looking fine. The compiling the Latex file on my PC was successful.

I would like to note that a key point in the first approach is the use of infinite-dimensional Grassmann-Banach algebras (or more general Banach Z_2-graded superalgebras.). (If I am not mistaken the Banach structure was used by B. DeWitt implicitly.) The Banach norm is necessary for definition of super-derivatives of functions (and not only). The normed version of DeWitt's approach was explored in papers by A. Rogers, V.S. Vladimirov, I.V. Volovich, A.Yu. Khrennikov etc.

During the process of compiling the Latex file, a problem arose that I could not solve. I will be grateful if somebody will help me. The trouble occurs for equation (15).

Thank a lot! I was close to proof of the second part, and planed to correct slightly the first one.

Yes. it's all about the zeros of the cubic G:=F' .

I am sorry. I have edited the Question: "extremum point(s)"' is replaced by "stationary point(s)".

Dear Toni, thank you so much! You have solved a half of the problem for the case i) $F'(1) \geq 0$ by proving that there are no roots of the cubic equation satisfying $x > 1$ for this case. But the case ii) $F'(1) < 0$ should be also considered in detail, since the last paragraph of my text is just an observation coming from numerical analysis, no more. Indeed, in the second case due to $F'(1) < 0$ and $F'(+ \infty) = + \infty$ there should exist at least one real root obeying $x > 1$. But a priori there may two roots or three. So, the uniqueness of such root should be proved.