Given the function:
[tex]f(x)=4-x^2[/tex]For the given function, we will determine the concavity between x = -1 and x = 5
By the average rate of change over the interval 3 ≤ x < 5
We will use the following formula:
[tex]\frac{f(5)-f(3)}{(5)-(3)}[/tex]First, we will find the value of f(5) and f(3)
[tex]\begin{gathered} x=5\rightarrow f(5)=4-5^2=-21 \\ x=3\rightarrow f(3)=4-3^2=-5 \end{gathered}[/tex]Substitute into the formula:
So, the average rate of change will be as follows:
[tex]\frac{f(5)-f(3)}{(5)-(3)}=\frac{(-21)-(-5)}{5-3}=\frac{-16}{2}=-8[/tex]As the average rate of change is negative, the concavity of the graph will be concave down
