Question

1. over which interval is the total change of the function g(x) least? 9) f(x) is increasing on (0,3) u (7,10) and decreasing on (3,7). g(x) is concave down, with a maximum at x = 5. on which interval in (0,10) is the function h(x)=f(x)+g(x) guaranteed to be decreasing? interval average roc 1 0,4 -7 4,5 3 5,8 -4 8,10

Explanation:

Step1: Recall the concept of total - change

The total change of a function over an interval $[a,b]$ is given by $g(b)-g(a)$. We know that the average rate of change (ROC) over an interval $[a,b]$ is $\text{ROC}=\frac{g(b)-g(a)}{b - a}$. So, the total change $g(b)-g(a)=\text{ROC}\times(b - a)$.

Step2: Calculate the total change for each interval

For the interval $[0,4]$: $\text{ROC} = 1$, $b - a=4 - 0=4$, so the total change is $1\times4 = 4$.
For the interval $[4,5]$: $\text{ROC}=-7$, $b - a=5 - 4 = 1$, so the total change is $-7\times1=-7$.
For the interval $[5,8]$: $\text{ROC}=3$, $b - a=8 - 5 = 3$, so the total change is $3\times3 = 9$.
For the interval $[8,10]$: $\text{ROC}=-4$, $b - a=10 - 8 = 2$, so the total change is $-4\times2=-8$.

Step3: Find the total change over all intervals

The total change of $g(x)$ over the entire domain is the sum of the total - changes over each sub - interval. Let $T$ be the total change. Then $T=4+( - 7)+9+( - 8)=4 - 7+9 - 8=-2$.
The total change over each interval:

• Interval $[0,4]$: Total change $=1\times(4 - 0)=4$.
• Interval $[4,5]$: Total change $=-7\times(5 - 4)=-7$.
• Interval $[5,8]$: Total change $=3\times(8 - 5)=9$.
• Interval $[8,10]$: Total change $=-4\times(10 - 8)=-8$.

The least total change occurs over the interval $[4,5]$ with a total change of $-7$.

For the second part (question 9):
We know that $h(x)=f(x)+g(x)$. The derivative of $h(x)$ is $h^\prime(x)=f^\prime(x)+g^\prime(x)$.
$f(x)$ is increasing on $(0,3)\cup(7,10)$ and decreasing on $(3,7)$, and $g(x)$ is concave - down with a maximum at $x = 5$.
The derivative of $g(x)$ is positive for $x\lt5$ and negative for $x\gt5$.
We want to find where $h^\prime(x)=f^\prime(x)+g^\prime(x)\lt0$.
On the interval $(3,5)$: $f(x)$ is decreasing ($f^\prime(x)\lt0$) and $g(x)$ is still increasing ($g^\prime(x)\gt0$), but we don't know the magnitudes.
On the interval $(5,7)$: $f(x)$ is decreasing ($f^\prime(x)\lt0$) and $g(x)$ is decreasing ($g^\prime(x)\lt0$). So, $h^\prime(x)=f^\prime(x)+g^\prime(x)\lt0$ on the interval $(5,7)$.

Answer:

1. The interval where the total change of $g(x)$ is least is $[4,5]$.
2. The interval where $h(x)$ is guaranteed to be decreasing is $(5,7)$.