Find all the points of local maxima and minima of the function. f(x) = x³ – 6x ² + 9x – 8.

Let y = f (x) = x³ – 6x² + 9 x – 8 Then,

=  = 3x² – 12x + 9 = 3(x² – 4x +3)

For a local maximum or local minimum, we have

= 0 3(x² – 4x + 3) = 0 x= 1, 3.

We have to examine whether these points are points of local maximum or local minimum or neither of them.

We have, = 3(x – 1) (x – 3)

 

Clearly,  changes sign from positive to negative as increases through 1.

  x = 1 is a point of local maximum.

Also,  changes sign from negative to positive as x increases through 3.

So x = 3 is a point of local minimum.