4. over which interval does ( y(x) ) have zero average rate of change?\n1) ( 4 leq x leq 10 )\n2) ( -5 leq x…

Explanation:

Step1: Recall the formula for average rate of change

The average rate of change of a function (y(x)) over the interval ([a,b]) is given by (\frac{y(b)-y(a)}{b - a}). For the average rate of change to be zero, (y(b)=y(a)).

Step2: Check each interval

• Interval (4\leq x\leq10): Let (a = 4) and (b=10). From the graph, (y(4)=0) and (y(10)\neq0). So (\frac{y(10)-y(4)}{10 - 4}\neq0).
• Interval (-5\leq x\leq1): Let (a=-5) and (b = 1). From the graph, (y(-5)=0) and (y(1)=0). Then (\frac{y(1)-y(-5)}{1-(-5)}=\frac{0 - 0}{6}=0).
• Interval (2\leq x\leq8): Let (a = 2) and (b=8). From the graph, (y(2)=3) and (y(8)\neq3). So (\frac{y(8)-y(2)}{8 - 2}\neq0).
• Interval (-10\leq x\leq-8): Let (a=-10) and (b=-8). From the graph, (y(-10)\neq y(-8)). So (\frac{y(-8)-y(-10)}{-8-(-10)}\neq0).

Answer:

(2)-5\leq x\leq1)