Answer:
5
Explanation:
To determine whether the function is increasing or decreasing over the interval [2, 7],we can analyze the sign of its derivative. The derivative of a function tells us how its values are changing at different points.
Given the function:
f(x)=-2x+4,5x+11,
if x<8
if x≥8
The function is defined by two different expressions for different ranges of
�
x. We will need to evaluate the derivatives of these two expressions separately and then compare the values at the interval boundaries (2 and 7) to determine whether the function is increasing or decreasing over the interval [2, 7].
Derivative of the first expression (-2x + 4):
Derivative of the second expression (5x + 11):
(x)=5
Now let's evaluate the derivatives at the interval boundaries:
2x=2:
For the first expression:
2f (2)=−2
For the second expression:
(2)=5f (2)=5
7x=7:
For the first expression:
(7)= −2f ′(7)=−2
For the second expression:
�
′
(
7
)
=
5
f
′
(7)=5
Since the derivative values are consistent across both expressions, the function is continuous at
�
=
8
x=8, which means that there's no sudden change in the slope at that point.
The derivative
�
′
(
�
)
f
′
(x) is positive (5) throughout the interval [2, 7], which means the function is increasing over this interval.
The rate of change of the function over the interval [2, 7] is 5, which is the constant value of the derivative during this interval. This indicates that for each unit increase in
�
x, the corresponding change in the function's value is 5.