Let be a non-constant thrice differentiable function defined on such that and . If is the minimum number or roots of in the interval [0, 6], then the value of is

Question ID - 153400 :-

Let be a non-constant thrice differentiable function defined on such that and . If is the minimum number or roots of in the interval [0, 6], then the value of is

1 Answer

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Answer Key : (6) -

(6)

(1)

On differentiating (1) w.r.t. , we get

(2)

Putting in (2), we get

Similarly

has minimum 7 roots in [0, 6]

Now, consider a function

As satisfy Rolle’s theorem in intervals and respectively

So, by Rolle’s theorem, the equation has minimum 6 roots

Now , where

Clearly has minimum 13 roots in [0, 6]

Hence again by Rolle’s theorem, has minimum 12 zeroes in [0, 6]

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