The number of values of k for which the equation x3−3x+k=0 has two distinct roots lying in the interval (0,1) are

The number of values of k for which the equation x33x+k=0 has two distinct roots lying in the interval (0,1) are

  1. A
  2. B
  3. C
  4. D

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    Solution:

    Let there be a value of k for which x33x+k=0 

    has two distinct roots between 0 and 1.

    Let a,b be two distinct roots of x33x+k=0 lying between 0 and 1 such that a<b.

    Let f(x)=x33x+k.Then f(a)=f(b)=0

    Between any two roots of a polynomial f (x) there exists at least one roots of its derivative f' (x). Therefore, f'(x)=3 x2-3 has at least one root between a and b. But, f'(x)=0 has two roots equal to± 1 which do not lie between a and b. Hence, f(x)=0 has no real roots lying between 0 and 1 for any value of k. 

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