CBSE Class 12-science Answered
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 | 04 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)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 = = –
Answered by | 04 Jun, 2014, 03:23: PM
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