QUESTION: x), points P and Q are at consecutive lowest and highest points with P occuring before Q. Find the slope of the line which passes through P and Q.
f(x) is at its lowest when sin(6πx) = -1, which is when the argument, 6πx = -π/2 + 2nπ.
i.e. 6x = -1/2 + 2n
x1 = -1/12 + n/3
f(x) is at its highest when sin(6πx) = 1, which is when the argument, 6πx = π/2 + 2nπ.
i.e. 6x = 1/2 + 2n
x2 = 1/12 + n/3
x1 and x2 are consecutive values for n = 0.
i.e. x1 = -1/12, x2= 1/12
f1 = f(x1) = -6, f2 = f(1/12) = 6
Our lowest and highest points now are (x1,f1) = (-1/12, -6), (x2,f2) = (1/12, 6).
Slope between these two points is given by m = (f2 - f1) / (x2 - x1)
m = (6 - (-6)) / (1/12 - (-1/12)) = 12/(1/6) = 72
Slope = 72