find the x-coordinate of all points of the graph of 3x-cos(2x) at which the tangent is perpendicular to the line 2x+4y=5
To find the x-coordinate of the points on the graph of 3x - cos(2x) where the tangent is perpendicular to the line 2x + 4y = 5, we need to find the derivative of 3x - cos(2x) and solve for x when the derivative is equal to -1/2 (the negative reciprocal of the slope of the line).
First, let's find the derivative of 3x - cos(2x). The derivative of 3x is 3, and the derivative of cos(2x) is -2sin(2x) by the chain rule. Therefore, the derivative of 3x - cos(2x) is:
d/dx (3x - cos(2x)) = 3 - (-2sin(2x)) = 3 + 2sin(2x)
Next, we set this derivative equal to -1/2 and solve for x:
3 + 2sin(2x) = -1/2
Subtracting 3 from both sides, we have:
2sin(2x) = -1/2 - 3
2sin(2x) = -7/2
Dividing both sides by 2, we get:
sin(2x) = -7/4
To solve for x, we need to find the inverse sine (sin^-1) of both sides:
2x = sin^-1(-7/4)
Dividing both sides by 2, we have:
x = (1/2)sin^-1(-7/4)
Therefore, the x-coordinate of all points on the graph of 3x - cos(2x) where the tangent is perpendicular to the line 2x + 4y = 5 is x = (1/2)sin^-1(-7/4), where sin^-1 represents arcsin or inverse sine.