Compare a cubic approximation with the actual function

$f$ crosses the origin, has a shallow negative minimum at $x=-\pi/4$, then rises to a maximum at $x=3\pi/4$ before falling towards its value at $3$. This follows from $f^{\prime}=e^x(\sin x+\cos x)$. The humped positive branch is $f$, not the polynomial.

$e^x\sin x=(1+x+x^2/2+\cdots)(x-x^3/6+\cdots)=x+x^2+x^3/3+\cdots$. Thus $g(x)=x+x^2+x^3/3$.

$g^{\prime}(x)=(x+1)^2\ge0$, so $g$ is increasing with a stationary inflection at $(-1,-1/3)$. It reaches $g(3)=21$, while $f$ turns down after $3\pi/4$. The source drawing swaps its f/g labels; the public reconstruction plots the actual formulas and corrects those labels. The original source drawing remains unchanged in the evidence archive.

Let $E(x)=f(x)-g(x)$. Solving $E(a)=0.5$ and $E(b)=-0.5$ in the given domain gives $a\approx-1.96187746881$, $b\approx1.55969700385$. The strict condition $|E(x)|<0.5$ holds for $a<x<b$; if “within” includes equality, $|E(x)|\le0.5$ holds for $a\le x\le b$.

The source reports the rounded open interval $-1.96<x<1.56$. Its displayed error expression mistakenly uses $x^3/3!$, although part (ii) defines $g$ with $x^3/3$; the numerical bounds correspond to the correct cubic. Use the unrounded roots when classifying a value very close to a boundary.