Question

Find an equation of a line that is tangent to y=2sinx and whose slope is a maximum

Answer 1

The slope is a max/min when the second derivative is zero
y' = 2cosx
y'' = -2sinx

-2sinx = 0
sinx = 0
x = 0, π, 2π , ..
when x=0, y' = 2cos0 = 2 , then slope = 2
when x = π, y' = 2cosπ = -2

let's use x=0, then y = 0
slope =2

y = 2x + b
at (0,0)
0 = 0+b

A tangent equation with a maximum slope is
y = 2x