The Function f (x) = x (x2 – 4)n (x2 – x + 1), n Ïµ N assumes A Local Minima At x = 2, Then

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Question

The function f (x) = x (x² – 4)ⁿ (x² – x + 1), n Ïµ N assumes a local minima at x = 2, then

medium

Solution

Correct option is

‘n’ can be any even number

When we differentiate it less than n times there will be a factor or (x – 2) in every for n₁.

If we differentiate in n times, only term, with no factor of (x – 2) is one is which (x – 2)n is differention n times

So if x = 2 is a minima, n must be even.

OR

Critical points: x = 2, 0, –2.

At x = 2; we have a local extremes if

x = 2 is a double or ever root ⇒ x Ïµ even.

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