graph of f\nthe graph of the function f, consisting of two line segments, is shown in the figure above. let…

graph of f\nthe graph of the function f, consisting of two line segments, is shown in the figure above. let g be the function given by ( g ( x ) = 2 x + 1 ), and let h be the function given by ( h ( x ) = f ( g ( x ) ) ). what is the value of ( h ^ { prime } ( 1 ) )?\na -4\nb -2\nc 4\nd 6

graph of f\nthe graph of the function f, consisting of two line segments, is shown in the figure above. let g be the function given by ( g ( x ) = 2 x + 1 ), and let h be the function given by ( h ( x ) = f ( g ( x ) ) ). what is the value of ( h ^ { prime } ( 1 ) )?\na -4\nb -2\nc 4\nd 6

Answer

Explanation:

Step1: Apply the chain rule

The chain rule states that (h^{\prime}(x)=f^{\prime}(g(x))\cdot g^{\prime}(x)). First, find (g^{\prime}(x)). Since (g(x) = 2x+1), then (g^{\prime}(x)=2).

Step2: Find (g(1))

Substitute (x = 1) into (g(x)): (g(1)=2\times1 + 1=3).

Step3: Find (f^{\prime}(3))

The function (f(x)) has two line - segments. For (x\in[0,1]), the slope (m_1=\frac{4 - 1}{1-0}=3). For (x\in[1,4]), the slope (m_2=\frac{-2 - 4}{4 - 1}=\frac{-6}{3}=-2). Since (g(1) = 3) and (3\in[1,4]), (f^{\prime}(3)=-2).

Step4: Calculate (h^{\prime}(1))

Substitute into the chain - rule formula: (h^{\prime}(1)=f^{\prime}(g(1))\cdot g^{\prime}(1)). We know that (g^{\prime}(1) = 2) and (f^{\prime}(g(1))=f^{\prime}(3)=-2). So (h^{\prime}(1)=(-2)\times2=-4).

Answer:

A. - 4