Parallel Tangents To A Curve
The following is the plot of
. Points
,
,
and
have abscissae
,
,
and
, respectively. The line tangent to the graph at one of these points is parallel to the tangent at another point. Determine the pair(s) that satisfy(ies) this property.
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1 solution
If (a, f(a)) and (b, f(b)) are two points on any continuous, differentiable curve f(x), then the slopes of the tangent lines drawn at x = a and x = b are parallel iff f' (a) = f' (b). If we have f(x) = x + cos(x), then f' (x) = 1 - sin(x). Clearly points B and C above have parallel tangents since 1 - sin(pi) = 1 - sin(2*pi) = 1. However, sin(2) > sin(8) which implies 1 - sin(2) < 1 - sin(8) and non-parallel tangents at points A and D.