use the figure to find the indicated derivatives, if they exist. (if an answer not exist, enter dne.) let…

Explanation:

Step1: Recall sum - rule of derivatives

The sum - rule states that if $h(x)=f(x)+g(x)$, then $h^{\prime}(x)=f^{\prime}(x)+g^{\prime}(x)$. So, $h^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)$.

Step2: Find $f^{\prime}(2)$

The function $f(x)$ is a straight - line. The slope of the line $f(x)$ can be found using two points. Let's use $(0,4)$ and $(4,1)$. The slope formula is $m=\frac{y_2 - y_1}{x_2 - x_1}$. So, $f^{\prime}(x)=\frac{1 - 4}{4-0}=-\frac{3}{4}$. Then $f^{\prime}(2)=-\frac{3}{4}$.

Step3: Find $g^{\prime}(2)$

The function $g(x)$ is a piece - wise linear function. For $x\geq3$, the slope of $g(x)$ is $1$ (since it is a line with a slope of $1$ for $x\geq3$). For $x < 3$, we consider the part of the line from $(0,0)$ to $(3,1)$. The slope of this line is $m=\frac{1 - 0}{3 - 0}=\frac{1}{3}$. Since $2<3$, $g^{\prime}(2)=\frac{1}{3}$.

Step4: Calculate $h^{\prime}(2)$

$h^{\prime}(2)=f^{\prime}(2)+g^{\prime}(2)=-\frac{3}{4}+\frac{1}{3}$. We find a common denominator, which is $12$. Then $h^{\prime}(2)=\frac{-9 + 4}{12}=-\frac{5}{12}$.

Answer:

$-\frac{5}{12}$