which interval has a positive average rate of change for $f(x)=-x^{2}+2x + 5$?\na. $1,2$\nb. $0,3$\nc…

which interval has a positive average rate of change for $f(x)=-x^{2}+2x + 5$?\na. $1,2$\nb. $0,3$\nc. $2,3$\nd. $0,1$

Answer

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}).

For interval ([1,2]):
- First, find (f(1)) and (f(2)).
- (f(x)=-x^{2}+2x + 5), so (f(1)=-(1)^{2}+2\times(1)+5=-1 + 2+5 = 6).
- (f(2)=-(2)^{2}+2\times(2)+5=-4 + 4+5 = 5).
- Then, the average rate of change is (\frac{f(2)-f(1)}{2 - 1}=\frac{5 - 6}{1}=-1).
For interval ([0,3]):
- Find (f(0)) and (f(3)).
- (f(0)=-(0)^{2}+2\times(0)+5 = 5).
- (f(3)=-(3)^{2}+2\times(3)+5=-9+6 + 5 = 2).
- The average rate of change is (\frac{f(3)-f(0)}{3 - 0}=\frac{2 - 5}{3}=-1).
For interval ([2,3]):
- (f(2) = 5) (calculated above), (f(3)=2).
- The average rate of change is (\frac{f(3)-f(2)}{3 - 2}=\frac{2 - 5}{1}=-3).
For interval ([0,1]):
- (f(0) = 5) (calculated above), (f(1)=6).
- The average rate of change is (\frac{f(1)-f(0)}{1 - 0}=\frac{6 - 5}{1}=1).

d. ([0,1])