How to solve it: show truning points and ppoints of function and max & min ?

Question

y=x^3-6x^2+9x-5

please draw the graph too

Answer 1

y = x^3 - 6x^2 + 9x - 5

The derivative is y' = 3x^2 -12x +9

Set y' = 0.

3x^2 -12x +9 = 0

Factor.

3(x^2-4x+3) = 0

3(x-1)(x-3) = 0

x = 1 or x = 3 are the turning points.

You can use the second derivative to determine local max/min.

y'' = 6x -12

For x = 1, y'' = 6(1)-12 = -6 < 0, so this is a local max.

For x = 3, y'' = 6(3)-12 = 6 > 0, so this is a local min.

To find the points, sub each into the original equation.

For x = 1, y = (1)^3 -6(1)^2 +9(1)-5 = 1-6+9-5 = -1

For x = 3, y = (3)^3 -6(3)^2 +9(3)-5 = 27-54+18-5 = -14

(1,-1) is a local max.

(3,-14) is a local min.