Question

if ( f(x)=3x^{2}-7x + 4 ), find ( f(-5) ).
use this to find the equation of the tangent line to the parabola ( y = 3x^{2}-7x + 4 ) at the point ( (-5,114) ). the equation of this tangent line can be written in the form ( y=mx + b )
where ( m ) is:
and where ( b ) is:
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Explanation:

Step1: Find the derivative of \(f(x)\)

Using the power rule \((x^n)^\prime = nx^{n - 1}\), for \(f(x)=3x^{2}-7x + 4\), we have \(f^\prime(x)=(3x^{2})^\prime-(7x)^\prime+(4)^\prime\).
\(f^\prime(x)=3\times2x-7\times1+0 = 6x-7\).

Step2: Calculate \(f^\prime(-5)\)

Substitute \(x = - 5\) into \(f^\prime(x)\).
\(f^\prime(-5)=6\times(-5)-7=-30 - 7=-37\). So \(m=-37\).

Step3: Find the equation of the tangent line \(y = mx + b\)

We know the point \((x_0,y_0)=(-5,114)\) and \(m=-37\). Substitute into \(y=mx + b\) (i.e., \(y_0=mx_0 + b\)).
\(114=-37\times(-5)+b\).
\(114 = 185 + b\).
Solve for \(b\): \(b=114 - 185=-71\).

Answer:

\(f^\prime(-5)=-37\), \(m=-37\), \(b=-71\)