Find all the points of local maxima and local minima as well as the corresponding local maximum and local values for the function f (x) = (x –1)³ (x + 1)²

Let y = f (x) = (x – 1)³(x + 1)² .Then,

= 3(x – 1)² (x + 1)² + 2 (x + 1) (x – 1)³

 = (x–1)² (x + 1) {3(x + 1) + 2(x – 1)}

= (x – 1)² (x + 1) (5x + 1)

For a local maximum or local minimum, we have

 = 0 (x – 1)² (x + 1) (5x + 1) = 0

x = 1 or, x = –1or, x = –

Now, we have to examine whether these points are points of local maximum or local minimum or neither of them.

Since (x – 1)²  is always positive, therefore the sign of  is same as that the ( x – 1) (5x + 1)

 

Clearly,  does not change its sign as x passes through 1. So x = 1 is neither a point of local maximum nor a point of local minimum. In fact, x = 1is a point of inflexion.

Clearly,  changes sign from positive to negative as x passes through – 1.

So, x = – 1 is a point of local maximum.

The local maximum value of f (x) at x = –1 is f (– 1) = (– 2)³  (– 1 + 1)²  = 0

It is evident from Fig. 18.18 that  Changes sign negative to positive as x passes through

So, x =  is a point of local minimum

The local minimum value of f (x) at x = –  is

f = = –