Question

the derivative of the function ( f ) is given by ( f^{prime}(x)=-3x + 4 ) for all ( x ), and ( f(-1)=6 ). which of the following is an equation of the line tangent to the graph of ( f ) at ( x=-1 )?
a ( y=-3x + 3 )
b ( y=-3x + 4 )
c ( y=7x + 6 )
d ( y=7x + 13 )

Explanation:

Step1: Find the slope of the tangent line

The slope \(m\) of the tangent line to the graph of \(y = f(x)\) at \(x=a\) is \(f^{\prime}(a)\).
Given \(f^{\prime}(x)=-3x + 4\) and \(a=-1\), then \(m=f^{\prime}(-1)=-3\times(-1)+4=3 + 4=7\).

Step2: Find the point on the function

We know that when \(x=-1\), \(y = f(-1)=6\). So the point \((x_0,y_0)=(-1,6)\).

Step3: Use the point - slope form of a line

The point - slope form of a line is \(y - y_0=m(x - x_0)\).
Substitute \(m = 7\), \(x_0=-1\), and \(y_0 = 6\) into the formula:
\(y-6=7(x + 1)\).
Expand the right - hand side: \(y-6=7x+7\).
Add 6 to both sides: \(y=7x+13\).

Answer:

D. \(y = 7x+13\)