Science:Math Exam Resources/Courses/MATH100 A/December 2023/Question 27(b)/Solution 1

We analyze the sign of $f'(x)=-4e^{-x/2}(x-2)$. Note that $f'(x)=ABC$, where $A=-4$, $B=e^{-x/2}$ and $C=x-2$. A product of 3 numbers is positive if the numbers are either all positive or if two of them are negative. Since $A$ is negative for all $x$, it follows that $f'(x)$ is positive if and only if precisely one of $B$ or $C$ is negative. $B$ is positive for all $x$, so we only need to analyze the sign of $C$, which is clearly positive for $x<2$ and negative for $x>2$.

Putting it all together, we find that $f'(x)$ is positive if $x\in [0,2)$ and negative if $x\in (2,\infty )$, so $f$ is increasing on the interval $[0,2)$ and decreasing on the interval $(2,\infty )$.