I'm trying to determine the x value(s) at which a function has horizontal tangents. The function is: (sin 4x)/(e^(4x)) at which -2<x<2
For the derivative I got: ((4(cos(4x))(e^(4x)))-((4e^(4x))(sin(4x))))/(e^((4x))
that denominator in the derivative is e to the 4x squared. I also apologize for all of the parenthesis. Assuming that the derivative is correct, I set it equal to zero and solve for x right? Well I did that and I couldn't really figure out how to get x by itself. I got the solving down to: (4e^(4x))((cos (4x)) - (sin (4x)))
obviously (4e^(4x)) will never equal zero because a zero in the exponent would make it 1, bit I cannot figure out a way to get x alone with cos(4x) - sin(4x) Am I doing something wrong?
I would have just noted that sin(4x)- cos(4x)= 0 is the same as sin(4x)= cos(4x). For what angles are sine cosine the same? Of course, that is the same as saying sin(4x)/cos(4x)= tan(4x)= 1. That is why it is important that tangent is positive in the first and third quadrants.