Move the pink slider to reposition the blue secant line.

Move the green point to reposition the red tangent line.

Observe there is no tangent line at the red point of inflection.

Show Auxiliary graphs.

The red tangent line is always parallel to the  blue secant line.

dist = 1
f(x) = (((x - 1) * (x - 5)) * (x - 10))
A = Point(f)
a = Tangent(A, f)
C: (x(A) + dist, a(x(A) + dist))
B: (x(A) - dist, a(x(A) - dist))
b = Segment(B, C)
slope = Slope(a)
c: x(A)
d(n) = 16 - (3 * c) - n
mnpos(n) = ((-d(n)) + sqrt(d(n)^(2) - (4 * (n^(2) + ((3 * c) * n) - (16 * n))))) / 2
nslider = 1.1 (slider from 0 to 2.4, step 0.1)
m: mnpos(nslider)
g(x) = x^(3) - (16 * x^(2)) + (65 * x) - 50
D: (c - m, f(c - m))
E: (c + nslider, f(c + nslider))
mnneg(n) = ((-d(n)) - sqrt(d(n)^(2) - (4 * (n^(2) + ((3 * c) * n) - (16 * n))))) / 2
r: mnneg(nslider)
F: (c - r, f(c - r))
e = Line(D, E)
h = Segment(D, E)
i = Segment(E, F)
p = TurningPoint(f)
j = false