a. find the derivative of f at x. that is, fi...

# Explanation: ## Step1: Recall derivative property The derivative of a linear - function $y = ax + b$ is $y'=a$. For $f(x)=-3x + 1$, by the power - rule of differentiation $\frac{d}{dx}(x^n)=nx^{n - 1}$, where for $y=-3x+1$, the derivative of $-3x$ is $-3\times1\times x^{1 - 1}=-3$ and the derivative of the constant 1 is 0. So $f'(x)=-3$. ## Step2: Find slope at $x = - 2$ The slope of the tangent line to the graph of $y = f(x)$ at a point $x = c$ is given by $m=f'(c)$. Since $f'(x)=-3$ for all $x$, when $x=-2$, $m_{\tan}=f'(-2)=-3$. ## Step3: Find slope at $x = 4$ Since $f'(x)$ is a constant function equal to $-3$ for all $x$, when $x = 4$, $m_{\tan}=f'(4)=-3$. # Answer: b. $m_{\tan}=-3$ $m_{\tan}=-3$