Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 28(a)/Solution 1

Since $e^{x}>0$ for all $x$, it follows that the derivative, given by $f'(x)=-e^{-x}$, is negative for all $x$, and the second derivative, which is equal to $f$ itself is positive. Thus the graph of the function is decreasing, concave up and has no critical or inflection points. Taking the limit as $x\rightarrow \infty$ reveals that it has a horizontal asymptote $y=0$.

The plot for the function and the tangent line is given below. The area is that enclosed by the the triangle whose vertices are the origin and the two intersections of the tangent line with the x- and y-axes.

Function and tangent line.