#"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"#
