Science:Math Exam Resources/Courses/MATH104/December 2014/Question 05 (f)/Solution 1

We know from part $a$ that the critical point of $f$ occurs at $x=2$; hence the graph of the function will change from decreasing to the left of $x=2$ to increasing to the right of $x=2$ (see first image below). Next, we know that in the interval to the left of the vertical asymptote at $x=0$, the function is increasing. Moreover, in that interval, as $x$ tends to $-\infty$, $f(x)$ tends to $0$. From part $d$, we know that in this interval, the function is concave up (see second image below).

Next, we focus on the intervals to the right of the vertical asymptote. In the interval $(0,2)$, we recall that the function is decreasing whilst in the interval $(2,\infty )$, the function is increasing (see third image below). From part $d$, we know that it is concave up in $(0,\infty )$, and flies off to infinity as $x$ tends to $\infty$. Combining these facts we have the graph depicted in the fourth image below:

f(x) is decreasing to the left of x=2, and is increasing to the right of x=2

f(x) tends to 0 as x tends to negative infinity and is concave up; there is also a vertical asymptote at x=0

f(x) is increasing in (2, infinity)

Final plot of f(x)