At what exact point on the curve y = 6 + 2ex − 6x is the tangent line parallel to the line 6x − y = 7?

Question

At what exact point on the curve y = 6 + 2e^x − 6x

 is the tangent line parallel to the line 6x − y = 7?

Answer 1

6x - y = 7 So, y = 6x - 7. Slope = 6

y = 6 + 2e^x - 6x So, y' = 2e^x - 6.

We need to solve the equation 2e^x - 6 = 6

2e^x = 12

e^x = 6

x = ln6

When x = ln6, y = 6 + 2e^ln6 - 6ln6 = 6 + 2(6) - 6ln6 = 18 - 6ln6.

Point of tangency = (6, 18 - 6ln6)

Answer 2

For the tangent to be parallel to 6x-y=7, the tangent line has to be equal to the slope of the line. For this case, the slope of the line is m = 6.

y'(x)=2e^x-6 = m = 6

2e^x = 12

e^x=6

x = ln(6)

y(ln(6)) = 6+2e^ln(6) -6ln(6)

= 6(1-ln(6)) +2(6)

≈ 7.25

P = (ln(6), 7.25)