PLEASE ANSWER CORRECTLY
PLEASE ANSWER CORRECTLY The number of values of k for which the equation x3-3x+k=0 has two different roots lying in the interval (0, 1) are infinitely many no value of k satisfies the requirement.

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W e h a v e, f (x) = x^{3} - 3 x + k f' (x) = 3 x^{2} - 3 B u t, f' (x) = 0 \Rightarrow 3 x^{2} - 3 = 0 \Rightarrow x^{2} = \frac{3}{3} = 1 \Rightarrow x = + 1, - 1 t h e c o n d i t i o n o f t w o d i s t i n c t r o o t i s f (α) \times f (β) = 0 w h e r e α a n d β a r e r o o t s o f f' (x) = 0 \Rightarrow f (1) \times f (- 1) = (1^{3} - 3 \times 1 + k) \times ({(- 1)}^{3} - 3 \times (- 1) + k) = 0 \Rightarrow (1 - 3 + k) \times (- 1 + 3 + k) = 0 \Rightarrow (k - 2) \times (k + 2) = 0 \Rightarrow k = 2, - 2 t h u s k \in (- 2, - 1, 0, 1, 2 - 2) t h u s t h e n u m b e r o f v a l u e o f k i n (0, 1) i s 0, 1 i t s e l f i . e, 2 H e n c e o p t i o n (2) i s c o r r e c t

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​​PLEASE ANSWER CORRECTLY​​PLEASE ANSWER CORRECTLY The number of values of k for which the equation x3-3x+k=0 has two different roots lying in the interval (0, 1) are infinitely many no value of k satisfies the requirement.