At what point on the curve #y=1+2e^x-3x# is the tangent line parallel to the line #3x-y=5#?

1 Answer
Jun 8, 2017

#(ln3,7-3ln3)#

Explanation:

#3x-y=5#
#y=3x+5#

Here we have slope #m=3# since it is in the form #y=mx+b#.
Since the lines are parallel, they have the same slope.

Find the derivative to find the point wih tangent line of slope #3#:
#y=1+2e^x-3x#
#y'=0+2e^x-3#
#3=2e^x-3#
#3=e^x#
#x=ln3#

Find the #y#-coordinate.
#y=1+2e^x-3x#
#y=1+2e^(ln3)-3ln3#
#y=7-3ln3#

Thus it is parallel to the tangent line at the point #(ln3,7-3ln3)#.