What was your life's turning point?

Mathematics Question - Can a point of inflection be a turning point as well?

• For y= x^4 - 12x^3 + 48x^2 - 64x, I find that its turning points occur at: x=1 and x=4. I also find that its points of inflection occur at: x=2 and x=4. *Note, I did check properly that these points of inflection actually do occur by checking left and right of the x=2 and x=4 points by use of the second derivative test. *Notice the double up at x=4. Does this mean that a point of inflection can also act as a turning point or have I done something wrong. A straight answer and some mathematics to support your claim would be much appreciated, if you can :) Data from Wolfram: http://www.wolframalpha.com/input/?i=points+of+inflection+y%3D+x^4+-+12x^3+%2B+48x^2+-+64x%2C http://www.wolframalpha.com/input/?i=turning+points+y%3D+x^4+-+12x^3+%2B+48x^2+-+64x%2C

• Answer:

  global minimum is at (1 , -27) inflection points are at (2 , -16) & (4 , 0) Edit: y= x^4 - 12x^3 + 48x^2 - 64x y' = 4x^3 - 36x^2 + 96x - 64 = 0 4(x^3 - 9x^2 + 24x - 16) = 0 4(x - 1)(x - 4)^2 = 0 x = 1 , 4 y = -27 , 0 y" = 12x^2 - 72x + 96 y"(1) = 12 - 72 + 96 > 0 => (1 , -27) min y"(4) = 192 - 288 + 96 = 0 => inflection point(4 , 0) 12(x^2 - 6x + 8) = 0 (x - 2)(x - 4) = 0 x = 2 , 4 => inflection points.