the graphs of the piecewise linear functions f and g are shown above. if the function h is defined by…

Explanation:

Step1: Find the derivatives of (f(x)) and (g(x)) at (x = 2)

The slope of a linear function (y=mx + b) (where (m) is the slope) is used to find the derivative. For (y = f(x)): Using two points ((0,0)) and ((6,3)) on the non - vertical part of (y = f(x)) near (x = 2), the slope (m_f=\frac{3-0}{6 - 0}=\frac{1}{2}), so (f^{\prime}(2)=\frac{1}{2}) For (y = g(x)): Using two points ((3,0)) and ((6,3)) on the non - vertical part of (y = g(x)) near (x = 2), the slope (m_g=\frac{3-0}{6 - 3}=1), so (g^{\prime}(2)=1) Also, (f(2)=\frac{1}{2}\times2 = 1) and (g(2)=1\times(2 - 3)+0=- 1)

Step2: Use the product rule

The product rule states that if (h(x)=f(x)g(x)), then (h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)) Substitute (x = 2) into the product rule formula: (h^{\prime}(2)=f^{\prime}(2)g(2)+f(2)g^{\prime}(2)) (h^{\prime}(2)=\frac{1}{2}\times(-1)+1\times1) (h^{\prime}(2)=-\frac{1}{2}+1=\frac{1}{2})

Answer:

A. 2