Question

F(x)=x^2−x−1

Over which interval does f have an average rate of change of zero?

Choose 1 answer:

(Choice A)

−1≤x≤2

(Choice B)

−5≤x≤5

(Choice C)

2≤x≤3

(Choice D)

−3≤x≤−2

Answer 1

Step-by-step explanation:

x²-x-1

=x²-x+x-1

=x(x-1) 1(x-1)

=(x+1) (x-1)

1) x+1=0 2) x-1=0

x= - 1 x= 1

Choice A is right.

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Answer 2

Answer:

Choice A is the correct answer.

Step-by-step explanation:

Average rate of change of a function $f(x)$ over an interval $[a,b]$ is,

Average rate of change = $\frac{f(b)-f(a)}{b-a}$.

Consider the function as follows:

$f(x)=x^2-x-1$ . . . . . (1)

(A) -1 ≤ $x$ ≤ 2

Substitute the values $x=-1$ and $x=2$ in the function (1), we get

$f(-1)=(-1)^2-(-1)-1$

         $=1$

$f(2)=2^2-2-1$

       $=1$

Then, the average rate of change is,

= $\frac{f(2)-f(-1)}{2-(-1)}$

= $\frac{1-1}{3}$

= 0

Thus, choice (A) is correct.

(B) -5 ≤ $x$ ≤ 5

Substitute the values $x=-5$ and $x=5$ in the function (1), we get

$f(-5)=(-5)^2-(-5)-1$

         $=29$

$f(5)=5^2-5-1$

       $=19$

Then, the average rate of change is,

= $\frac{f(5)-f(-5)}{5-(-5)}$

= $\frac{19-29}{10}$

= -10/10

= -1

Thus, choice (B) is incorrect.

(C) 2 ≤ $x$ ≤ 3

Substitute the values $x=2$ and $x=3$ in the function (1), we get

$f(3)=(3)^2-3-1$

         $=5$

$f(2)=2^2-2-1$

       $=1$

Then, the average rate of change is,

= $\frac{f(3)-f(2)}{3-2}$

= $\frac{5-1}{1}$

= 4

Thus, choice (C) is incorrect.

(D) -3 ≤ $x$ ≤ -2

Substitute the values $x=-3$ and $x=-2$ in the function (1), we get

$f(-3)=(-3)^2-(-3)-1$

         $=11$

$f(-2)=(-2)^2-(-2)-1$

       $=5$

Then, the average rate of change is,

= $\frac{f(-2)-f(-3)}{-2-(-3)}$

= $\frac{5-11}{1}$

= -6

Thus, choice (D) is incorrect.

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