what is the average rate of change of the function f(x) = x² - 5x + 14 on the interval from x = 3 to x = 5?

Answer

Explanation:

Step1: Recall average - rate - of - change formula

The average rate of change of a function $y = f(x)$ on the interval $[a,b]$ is $\frac{f(b)-f(a)}{b - a}$. Here, $a = 3$, $b = 5$, and $f(x)=x^{2}-5x + 14$.

Step2: Calculate $f(5)$

Substitute $x = 5$ into $f(x)$: $f(5)=5^{2}-5\times5 + 14=25-25 + 14=14$.

Step3: Calculate $f(3)$

Substitute $x = 3$ into $f(x)$: $f(3)=3^{2}-5\times3 + 14=9-15 + 14=8$.

Step4: Calculate the average rate of change

$\frac{f(5)-f(3)}{5 - 3}=\frac{14 - 8}{2}=\frac{6}{2}=3$.