Question

find all points (x,y) on the graph of f(x)=3x^2 - 4x with tangent lines parallel to the line y = 14x + 1. the point(s) is/are . (type an ordered pair. use a comma to separate answers as neeeded.)

Explanation:

Step1: Find the derivative of f(x)

The derivative of $f(x)=3x^{2}-4x$ using the power - rule $(x^n)' = nx^{n - 1}$ is $f'(x)=6x - 4$.

Step2: Determine the slope of the given line

The line $y = 14x+1$ has a slope $m = 14$.

Step3: Set the derivative equal to the slope

Since tangent lines are parallel to the given line, we set $f'(x)=14$. So, $6x - 4=14$.

Step4: Solve for x

Add 4 to both sides: $6x=14 + 4=18$. Then divide by 6: $x = 3$.

Step5: Find the y - value

Substitute $x = 3$ into $f(x)=3x^{2}-4x$. So, $y=f(3)=3\times3^{2}-4\times3=3\times9 - 12=27-12 = 15$.

Answer:

$(3,15)$