= Solution
The <absolute value> in the original PDF reflects every negative <sine> arch upward. The graph is nonnegative, with zeros at $x=k\pi$ for $k=0,1,2,3,4$, and it lies below the decaying envelope $e^{-x}$. On the $k$th arch, <differentiation> gives a factor proportional to $\cos x-\sin x$. Thus its unique <local maximum> is
$$
x_k=\frac\pi4+k\pi,\qquad
f(x_k)=\frac{e^{-(\pi/4+k\pi)}}{\sqrt2},\qquad k=0,1,2,3.
$$
Successive maximum heights have ratio $e^{-\pi}$. At each interior zero the one-sided slopes have opposite signs, producing a corner. The smaller panels below magnify the later arches so that all four remain visible.
\Image[/past-exam-of-the-mathematics-course-of-the-year-1-of-the-natural-sciences-course-of-the-university-of-cambridge/2012/paper-1-damped-absolute-sine.png]
{title=Four positive exponentially damped sine arches, with separate linear-scale enlargements of the later arches}
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