Question

Determine the concavity of the graph of f(x) = 4 - x^2 between x= -1 and x = 5 by calculating average rates of change over intervals of length 2. 1. The average rate of change over the interval 3 ≤ 2 < 5 =

Answer 1

Given the function:

`f(x)=4-x^2`

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:

`(f(5)-f(3))/((5)-(3))`

First, we will find the value of f(5) and f(3)

`\begin{gathered} x=5\rightarrow f(5)=4-5^2=-21 \\ x=3\rightarrow f(3)=4-3^2=-5 \end{gathered}`

Substitute into the formula:

So, the average rate of change will be as follows:

`(f(5)-f(3))/((5)-(3))=((-21)-(-5))/(5-3)=(-16)/(2)=-8`

As the average rate of change is negative, the concavity of the graph will be concave down