#"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"#