At what point on the curve $ y = 1 + 2e^x - 3x $ is the tangent line parallel to the line $ 3x - y = 5? $ Illustrate with a sketch.
Hey, it's Claire. So when you married here have to note that parallel lines have equal slope. So we have three X minus y this equal to five. This becomes why is equal to three X minus five. So the slope is three. You have. Why is equal to one plus two e to the X minus three X. We're gonna differentiate. We get to eat the X minus three we have to solve. When this is equal to three. So X becomes equal to Ln of three. Access equal to Ellen of three. You have why is equal to one plus two e the L one of three minus three l one of three just equal to seven minus three. Ellen three. So there's a tension that l and three comma seven minus three Ellen of three. So the equation of the tangent becomes set. Why minus seven minus three. Ellen of three over X minus Ellen of three. Equal three. And why becomes equal to three X minus six? Well, end of three plus seven. We're going to make a sketch says why x So we have why, for one plus two. Eat the X minus three x Well, look, something like this. Then we have our tangent in red and then three x minus y equals five for our blue are green.