= Solution
Assume the given <wavefunction> is zero outside $[0,L]$. <Wavefunction normalization> requires
$$
1=\int_0^L|Nx|^2dx=|N|^2\frac{L^3}{3},\qquad
\boxed{|N|=\sqrt{\frac3{L^3}}.}
$$
An arbitrary overall <quantum phase> is unobservable, so choose the positive real representative $N=\sqrt3/L^{3/2}$.
The <linear ramp state in an infinite square well> has a boundary subtlety: the limiting value at $L$ is $\sqrt{3/L}$, rather than zero. The function is still a normalized element of the well's <Hilbert space> $L^2(0,L)$, so <position measurement> and energy-projection probabilities are meaningful. It is not in the domain of the Dirichlet <Hamiltonian>, nor is it a finite-energy box state. Altering the value at the single endpoint does not cure its nonzero boundary trace; a physical finite-energy preparation must smooth the function to zero near the wall. The following probabilities refer to the supplied idealized state.
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