the derivative of the function ( f ) is given by ( f^{prime}(x)=-3x + 4 ) for all ( x ), and ( f(-1)=6 )…

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 )?\na ( y=-3x + 3 )\nb ( y=-3x + 4 )\nc ( y=7x + 6 )\nd ( y=7x + 13 )

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 )?\na ( y=-3x + 3 )\nb ( y=-3x + 4 )\nc ( y=7x + 6 )\nd ( y=7x + 13 )

Answer

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)