For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even.

Question 1: Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?

Question 2: Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive?

Question 3: Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$?

Comment: It appears that $$\ln(2g(t))=\sum_{k=1}^\infty\frac{t^{2k}}{2n(2n-1)}.$$ If this is true, then the positive answer to

A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even.

Question 1: Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?

Question 2: Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive?

Question 3: Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$?

A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even.

Question 1: Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?

Question 2: Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive?

Question 3: Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$?

A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.

For $t\in(-1,1)$, let $$f(t):=\left(\frac{1+t}{1-t}\right)^{(1-t)/2}+\left(\frac{1-t}{1+t}\right)^{(1+t)/2}$$ and $$g(t):=\frac1{f(t)}.$$ Note that the functions $f$ and $g$ are even.

Question 1: Is it true that all the even-order derivatives $f^{(2k)}$ of $f$ at $0$ are negative, except for $k=0$ and $k=2$?

Question 2: Is it true that all the even-order derivatives $g^{(2k)}$ of $g$ at $0$ are positive?

Question 3: Is there a simple, explicit, and accurate upper bound on the even-order derivatives $g^{(2k)}$?

Comment: It appears that $$\ln(2g(t))=\sum_{k=1}^\infty\frac{t^{2k}}{2n(2n-1)}.$$ If this is true, then the positive answer to

A correct and complete answer to any one of these three questions will be considered as a correct and complete answer to this entire post.