f(x)=x^{2}-x - 1\nover which interval does f have an average rate of change of zero?\nchoose 1…

f(x)=x^{2}-x - 1\nover which interval does f have an average rate of change of zero?\nchoose 1 answer:\n-1≤x≤2\n2≤x≤3\n-5≤x≤5\n-3≤x≤-2

f(x)=x^{2}-x - 1\nover which interval does f have an average rate of change of zero?\nchoose 1 answer:\n-1≤x≤2\n2≤x≤3\n-5≤x≤5\n-3≤x≤-2

Answer

Explanation:

Step1: Recall the average rate of change formula

The average rate of change of a function (y = f(x)) over the interval ([a,b]) is given by (\frac{f(b)-f(a)}{b - a}). We want (\frac{f(b)-f(a)}{b - a}=0), which implies (f(b)=f(a)).

Step2: Calculate (f(x)) for each option

  • Option A: (-1\leq x\leq2)
    • Calculate (f(-1)): (f(-1)=(-1)^{2}-(-1)-1=1 + 1-1=1)
    • Calculate (f(2)): (f(2)=2^{2}-2 - 1=4-2 - 1=1)
    • Since (f(-1)=f(2) = 1), the average rate of change (\frac{f(2)-f(-1)}{2-(-1)}=\frac{1 - 1}{3}=0)
  • Option B: (2\leq x\leq3)
    • (f(2)=1) (calculated above)
    • (f(3)=3^{2}-3 - 1=9-3 - 1=5)
    • (\frac{f(3)-f(2)}{3 - 2}=\frac{5 - 1}{1}=4\neq0)
  • Option C: (-5\leq x\leq5)
    • (f(-5)=(-5)^{2}-(-5)-1=25 + 5-1=29)
    • (f(5)=5^{2}-5 - 1=25-5 - 1=19)
    • (\frac{f(5)-f(-5)}{5-(-5)}=\frac{19 - 29}{10}=-1\neq0)
  • Option D: (-3\leq x\leq-2)
    • (f(-3)=(-3)^{2}-(-3)-1=9 + 3-1=11)
    • (f(-2)=(-2)^{2}-(-2)-1=4 + 2-1=5)
    • (\frac{f(-2)-f(-3)}{-2-(-3)}=\frac{5 - 11}{1}=-6\neq0)

Answer:

A. (-1\leq x\leq2)