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

Asked by Topperlearning User | 4th Jun, 2014, 01:23: PM

Expert Answer:

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)(– 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 = = –

Answered by  | 4th Jun, 2014, 03:23: PM