"Given:"
f(x)=x^3-6x^2+9x+1
f'(x)=3x^2-12x+9
"Solving for x where f'(x)=0,"
3x^2-12x+9=0
"Dividing by 3"
x^2-4x+3=0
x^2-3x-1x+3=0
x(x-3)-1(x-3)=0
(x-1)(x-3)=0
x=1, x=3. "form the points where the slope is zero"
f'(x)=3x^2-12x+9
f''(x)=6x-12
"Solving for x where f''(x) is zero"
6x-12=0
6(x-2)=0
x-2=0
x=2. "form the point where the curvature is zero."
"The point where the curvature changes its sign"
x=2, "forms a point of inflexion"
x=1, "the curvature is",
6x-12=6xx1-12=6-12=-6
x=3, "the curvature is",
6x-12=6xx3-12=18-12=+6
"From x=1, to x=2, the curvature is negative"
" indicating concave down"
"From x=2, to x=3, the curvature is positive"
" indicating concave up"
The function takes the values as evaluated below
f(x)=x^3-6x^2+9x+1
f(1)=(1)^3-6(1)^2+9(1)+1=1-6+9+1=-5+10=5
f(2)=(2)^3-6(2)^2+9(2)+1=8-24+18+1=-16+19=3
f(3)=(3)^3-6(3)^2+9(3)+1=27-54+27+1=-27+28=1
Here, the critical points are
(1,5), "where the slope is zero"
" and curvature is negative, thus being a maximum"" representing concave down"
(3,1), "where the slope is zero"
" and curvature is positive, thus being a minimum ""representing concave up"
However, the point
(2,3), "where the curvature is zero"
" and curve is changing from concave down to concave up"
