Answer: The correct option is (C) [tex]-\dfrac{32}{\pi}.[/tex]
Step-by-step explanation: We are given to find the rate of change between the interval [tex]x=\dfrac{\pi}{4}[/tex] and [tex]x=\dfrac{\pi}{2}.[/tex]
We know that
if y = f(x) is a function of x, then the rate of change from x = a to x = b is given by
[tex]R=\dfrac{f(b)-f(a)}{b-a}.[/tex]
From the graph, we note that
[tex]y\left(\dfrac{\pi}{4}\right)=1,\\\\\\y\left(\dfrac{\pi}{2}\right)=-7.[/tex]
Therefore, the rate of change between the interval [tex]x=\dfrac{\pi}{4}[/tex] and [tex]x=\dfrac{\pi}{2}.[/tex] is given by
[tex]R=\dfrac{y\left(\frac{\pi}{2}\right)-y\left(\frac{\pi}{4}\right)}{\frac{\pi}{2}-\frac{\pi}{4}}\\\\\\\Rightarrow R=\dfrac{-7-1}{\frac{\pi}{4}}\\\\\\\Rightarrow R=-8\times \dfrac{4}{\pi}\\\\\\\Rightarrow R=-\dfrac{32}{\pi}.[/tex]
Thus, the required rate of change is [tex]-\dfrac{32}{\pi}.[/tex]
Option (C) is CORRECT.
