Question

if $f(x)=3x^{2}-3x + 6$, find $f(4)$. use this to find the equation of the tangent line to the parabola $y = 3x^{2}-3x + 6$ at the point $(4,42)$. the equation of this tangent line can be written in the form $y=mx + b$ where $m$ is: and where $b$ is:

Explanation:

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

Using the power - rule $(x^n)'=nx^{n - 1}$, if $f(x)=3x^{2}-3x + 6$, then $f'(x)=(3x^{2})'-(3x)'+(6)'$.
$f'(x)=3\times2x-3+0 = 6x-3$.

Step2: Calculate $f'(4)$

Substitute $x = 4$ into $f'(x)$.
$f'(4)=6\times4-3=24 - 3=21$. So the slope $m$ of the tangent line is 21.

Step3: Find the value of $b$

The equation of the tangent line is $y=mx + b$, and the line passes through the point $(4,42)$. Substitute $x = 4$, $y = 42$ and $m = 21$ into $y=mx + b$.
$42=21\times4+b$.
$42 = 84+b$.
Solve for $b$: $b=42 - 84=-42$.

Answer:

$m$ is: 21
$b$ is: -42