All Tasks - Asymptote

Let $ \displaystyle f(x)= \dfrac{\pi}{2} -b \arctan(1-2x)$ with $b \in \mathbb{R}$, and let $f'(x)$ and $f''(x)$ be the first and the second derivative of the function $f$. Knowing that $\displaystyle f\left(\dfrac{1}{2}\right) \times f'\left(\dfrac{1}{2}\right) + f''\left(\frac{1}{2}\right) = 4 \pi$, then the value of $b$ is:

Consider $f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. Let $f'(x)$ be the first derivative of function $f(x)$, then $f'\left(-\dfrac{1}{2}\right)$ is:

Let $f$ and $g$ be two differentiable functions in their domain, such that: $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ and $g(x)=\left( x^2+1\right) f(x)$. Considering that $g'(x)$ is the first derivative of function $g(x)$, then $g'\left(\dfrac{1}{2}\right)$ is:

Consider $f(x)= -2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$ and range $D'_f = \left] -\pi, \pi \right[$. The analytic expression of the inverse function of $f$, $f^{-1}(x)$, and $f^{-1}\left(-\dfrac{\pi}{3} \right) $ are, respectively,

Consider $f(x)= \dfrac{\pi}{2} -2 \arctan(1-2x)$ with domain $D_f=\mathbb{R}$ and let $f^{-1}(x)= \dfrac{1}{2} -\dfrac{1}{2} \tan\left(\dfrac{\pi}{4}-\dfrac{x}{2}\right)$ be the analytic expression of the inverse function whose domain is $D_f^{-1}=\left]-\dfrac{\pi}{2}, \dfrac{3\pi}{2}\right[$. The solution of the equation $f\left( \dfrac{1-2x}{2}\right) +\cot\left(\arctan(1)\right) +f^{-1}(0) = 1$ is:

Consider $f(x)=-2 \arctan\left(2x\right)$ with domain $D_f=\mathbb{R}$. The solution of the equation $2\ f\left(\dfrac{x}{2} \right) -\dfrac{2}{\pi}\ f\left(\dfrac{1}{2} \right)=1$ is: