Find the area bounded by the curve $y = x e^{–x}$ ; xy = 0 and x = c where c is the x-coordinate of the curve's inflection point. The answer is $1-3e^{-2}$
I tried to plot the curve in desmos.com. xy=0 is a straight line. The point of infection of the curve is x=1 and $\infty$.
$xy=0$ is not a straight line: it is the union of two straight lines, viz. the coordinate axes.
I cannot understand why you assert that $x=\infty$ gives an inflection point. Neither do I get $c=1$ as the $x$-coordinate of the curve's inflection point. (What is the second derivative of $xe^{-x}$?)
The region between the curve, $xy=0$ and $x=c$ is the set $\{(x,y):0\le x\le c,0\le y\le xe^{-x}\}$ and has area $$\int_0^c xe^{-x}\,dx=\left[-(x+1)e^{-x}\right]_0^c=1-(c+1)e^{-c}$$ which gives a further clue as to the correct value of $c$.
