Find points of a tangent line

Question

Find all point/s where the slope to the tangent line to g(x) = 2e^x - 6x is 12.

Answer 1

It is important to note that the derivative is the slope of the tangent line at a particular value of x.

Therefore, in this problem, the question is asking when the derivative of g(x) is equal to 12 -- g'(x) = 12.

Finding g'(x), making use of the memorization rule d/dx [ e^x ] = e^x , we see that:

g'(x) = 2e^x - 6

Setting g'(x) equal to 12 and solving, making use of the natural log, gives us:

2e^x - 6 = 12 (given)

2e^x = 18 (add 6 to both sides)

e^x = 9 (divide both sides by 2)

ln( e^x ) = ln(9) (take the log of both sides)

x = ln(9) (simplify using ln(e^x) = x)

This leaves us with the answer to our question: the point x = ln(9) is where the slope of the tangent line to g(x) = 2e^x -6x is 12.

Answer 2

f'(x)=2e^x-6

The problem asks when does that function equal 12.

2e^x-6=12

2e^x=18

e^x=9

ln(e^x) = ln(9)

xln(e)=ln(9)

x=ln(9)

Check it out here to see the tangent line to the point on f(x) where the x-coordinate (abscissa) is ln(9):

desmos.com/calculator/1nozuqyisg