given that\n$f(x)=x^{7}h(x)$\n$h(-1)=5$\n$h(-1)=8$\ncalculate $f(-1)$.\nhint: use the product rule and the…

Explanation:

Step1: Apply the product rule

The product rule states that if (f(x)=u(x)v(x)), then (f^{\prime}(x)=u^{\prime}(x)v(x)+u(x)v^{\prime}(x)). Here (u(x) = x^{7}) and (v(x)=h(x)). The power rule ((x^{n})^{\prime}=nx^{n - 1}) gives (u^{\prime}(x)=7x^{6}). So (f^{\prime}(x)=7x^{6}h(x)+x^{7}h^{\prime}(x)).

Step2: Substitute (x = - 1)

Substitute (x=-1), (h(-1) = 5) and (h^{\prime}(-1)=8) into (f^{\prime}(x)). [ \begin{align*} f^{\prime}(-1)&=7(-1)^{6}h(-1)+(-1)^{7}h^{\prime}(-1)\ &=7\times1\times5+(-1)\times8 \end{align*} ]

Step3: Calculate the value

[ \begin{align*} f^{\prime}(-1)&=35 - 8\ &=27 \end{align*} ]

Answer:

(27)