Author: Larry Rose

f(x)

f ' (x)

f '' (x)

B

Drag the point on f(x) and observe the behavior 
on f(x), f '(x), and f ''(x).

Increasing

Decreasing

Maximum

Minimum

Concave Up

Concave Down

f(x)

f '(x)

f ''(x)

Increasing

Increasing

Positive

Positive

Positive

Negative

Negative

Negative

Decreasing

Decreasing

Zero

Zero

Zero

Maximum

Maximum

Minimum

Minimum

Concave Up

Concave Up

Concave Down

Concave Down

f(x) = 1.5 * x^(4.0) + 3.0 * x^(3.0) - 2.0 * x^(2.0) - 2.0 * x + 1.0
A = Point(f)
a: x(A)
d_1 = Derivative(f(x))
d_2 = Derivative(d_1)
B: (a, d_1(a))
(a, f(a)): (a, d_1(a))
y'': d_2(a)
y': d_1(a)
C: (a, d_2(a))
y_1: f(a)
D = (4.1393082312540574, 18.6886518919744)
E = (4.1393082312540574, 16.343097318342764)
F = (8.911298570711518, 16.10045374176018)
G = (8.911298570711518, 18.446008315391815)
K = (3.5084349321393424, -3.796319538011618)
L = (3.5084349321393424, -6.546280072614224)
M = (9.525995631387394, -6.627161264808419)
N = (9.509819392948556, -3.9580819224000066)
d_3 = Derivative(d_2)
y''': d_3(a)
A_1 = (2.85, 16.1)
A_2 = (5.1, 16.1)
A_3 = (7.35, 16.1)
A_4 = (9.8, 16.1)
B_1 = (2.85, 11.1)
B_4 = (9.8, 11.1)
C_1 = (2.85, 6.1)
D_1 = (2.85, 1.1)
C_4 = (9.8, 6.1)
D_4 = (9.8, 1.1)
G_2 = (5.1, 1.1)
G_3 = (7.35, 1.1)
b = Segment(A_2, G_2)
c = Segment(A_3, G_3)
d = Segment(A_1, A_4)
e = Segment(B_1, B_4)
g = Segment(C_1, C_4)
h = Segment(D_1, D_4)
H = (2.3820687439716868, -2.2952381392438963)
I = (2.349716267094009, 1.3444155094948456)
J = (4.666168345359365, 0.94001)
O = (4.666168345359365, -2.37612)
epsilon = 0.15
d_4 = Derivative(d_3)
y'''': d_4(a)
i = Segment(A_1, D_1)
j = Segment(A_4, D_4)