Answer:
c
Step-by-step explanation:
A negative
A
R
C
ARCA, R, C is obtained when the function value at the beginning of the interval is more than the function value at the end of the interval. When the beginning value is less than the terminal value, the
A
R
C
ARCA, R, C is positive.
For instance, consider the interval
2
≤
t
≤
5
2≤t≤52, is less than or equal to, t, is less than or equal to, 5:
g
(
2
)
=
−
3
g(2)=−3g, left parenthesis, start color #6495ed, 2, end color #6495ed, right parenthesis, equals, start color #ff00af, minus, 3, end color #ff00af, and
g
(
5
)
=
2
g(5)=2g, left parenthesis, start color #6495ed, 5, end color #6495ed, right parenthesis, equals, start color #ff00af, 2, end color #ff00af.
Since
−
3
<
2
−3<2start color #ff00af, minus, 3, end color #ff00af, is less than, start color #ff00af, 2, end color #ff00af, the
A
R
C
ARCA, R, C over the interval
2
≤
t
≤
5
2≤t≤52, is less than or equal to, t, is less than or equal to, 5 is positive.
Now consider the interval
−
5
≤
t
≤
2
−5≤t≤2minus, 5, is less than or equal to, t, is less than or equal to, 2:
g
(
−
5
)
=
0
g(−5)=0g, left parenthesis, start color #6495ed, minus, 5, end color #6495ed, right parenthesis, equals, start color #ff00af, 0, end color #ff00af, and
g
(
2
)
=
−
3
g(2)=−3g, left parenthesis, start color #6495ed, 2, end color #6495ed, right parenthesis, equals, start color #ff00af, minus, 3, end color #ff00af.
Since it is true that
0
>
−
3
0>−3start color #ff00af, 0, end color #ff00af, is greater than, start color #ff00af, minus, 3, end color #ff00af, the
A
R
C
ARCA, R, C over the interval
−
5
≤
t
≤
2
−5≤ t≤2minus, 5, is less than or equal to, space, t, is less than or equal to, 2 is negative!
gg has a negative average rate of change over the interval
−
5
≤
t
≤
2
−5≤t≤2minus, 5, is less than or equal to, t, is less than or equal to, 2.
