Question

e. if ( h(x)=f(x)g(x) ), then ( h(6)= )

Explanation:

Step1: Find the slope of \(y = f(x)\)

The line \(y = f(x)\) passes through \((-4, -1)\) and \((6, 2)\).
The slope \(m_f=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2-(-1)}{6 - (-4)}=\frac{3}{10}\)

Step2: Find the slope of \(y = g(x)\)

The line \(y = g(x)\) passes through \((4, 1)\) and \((6, 2)\).
The slope \(m_g=\frac{y_2 - y_1}{x_2 - x_1}=\frac{2 - 1}{6 - 4}=\frac{1}{2}\)

Step3: Use the product - rule

The product - rule states that if \(h(x)=f(x)g(x)\), then \(h^{\prime}(x)=f^{\prime}(x)g(x)+f(x)g^{\prime}(x)\)
At \(x = 6\), \(f(6)=2\), \(g(6)=2\), \(f^{\prime}(6)=\frac{3}{10}\), \(g^{\prime}(6)=\frac{1}{2}\)
\(h^{\prime}(6)=f^{\prime}(6)g(6)+f(6)g^{\prime}(6)\)
Substitute the values: \(h^{\prime}(6)=\frac{3}{10}\times2+2\times\frac{1}{2}\)
\(h^{\prime}(6)=\frac{3}{5}+1\)
\(h^{\prime}(6)=\frac{3 + 5}{5}=\frac{8}{5}=1.6\)

Answer:

\(1.6\)