I have been struggling with this problem I cannot get the right answer. Please give me the steps to solve this problem.

Question

Let f(x)= (2x-9)(x^2 + 6)
(A) Find the equation of the line tangent to the graph of f(x) at x=3
(B) Find the values(s) of x where the tangent line is horizontal.

Answer 1

I would expand the function first, then find the derivative.
f(x) = 2x^3 + 12x - 9x^2 - 54
f '(x) = 6x^2 + 12 - 18x
when x = 3
f(3) = -3(15) = -45
f '(3) = 6(9) + 12 - 18(3) = 12

A) so you have a slope of 12 and a point (3,-45)
Use y = mx + b to find the tangent equation

B) set 6x^2 - 18x + 12 = 0
and solve
Hint: divide both sides by 6, then it factors

Answer 2

Identify the variable, constant, and coefficient of the expression: –10k + 15