List of top Signals and Systems Questions on Applications of Fourier Transform for continuous and discrete time signals

A continuous-time signal \(x(t)\) is defined as \(x(t)=0\) for \(|t|>1\), and \(x(t)=1-|t|\) for \(|t|\le 1\). Let its Fourier transform be \(X(\omega)=\int_{-\infty}^{\infty} x(t)e^{-j\omega t}\,dt\). The maximum magnitude of \(X(\omega)\) is \(\underline{\hspace{2cm}}\).
 

Let $f(t)$ be an even function. The Fourier transform is $F(\omega)=\int_{-\infty}^\infty f(t)e^{-j\omega t}dt$. Suppose $\frac{dF(\omega)}{d\omega} = -\omega F(\omega)$ for all $\omega$, and $F(0)=1$. Then