The interval that function f(x)=x³-9x has a positive average rate of change is [ -4, -1 ].
Option D) is the correct answer.
Over which interval does f have a positive rate of change?
To determine whether the average rate of change is positive, f(max) - f(min) must be greater than zero.
Given the function; f(x) = x³ - 9x
We determine f(max) - f(min)
At interval [ -2, 1 ]
(x³ - 9x) - ( x³ - 9x )
( (1)³ - 9(1) ) - ((-2)³ - 9(-2))= -8 - 10 = -18 NOT POSITIVE
At interval [ -2, 3 ]
(x³ - 9x) - ( x³ - 9x )
( (3)³ - 9(3) ) - ((-3)³ - 9(-3))= 0 - 0 = 0 NOT POSITIVE
At interval [ -1, 2 ]
(x³ - 9x) - ( x³ - 9x )
( (2)³ - 9(2) ) - ((-1)³ - 9(-1))= -10 - 8 = -18 NOT POSITIVE
At interval [ -4, -1 ]
(x³ - 9x) - ( x³ - 9x )
( (-1)³ - 9(-1) ) - ((-4)³ - 9(-4))= 8 - (-28) = 36 POSITIVE
Therefore, the interval that function f(x)=x³-9x has a positive average rate of change is [ -4, -1 ].
Option D) is the correct answer.
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