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Question

Find the range of values of the parameter a so that x3−3x+a=0 has three real and distinct roots.

A
(1,1)
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B
(2,2)
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C
(3,3)
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D
(4,4)
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Solution

The correct option is C (2,2)
Let f(x)=x33x+a
f(x)=3x23=3(x1)(x+1)
f(x)=0x=1,1
At x=1, f′′(x)>0, hence x=1 is a point of maxima and similarly x=1 is a point of minima.
For real roots, the maxima should have a value >0 and the minima should be <0.
Now, f(1)=a2, f(1)0=a+2
Hence,
a2<0 and a+2>0
Thus, the given equation would have real and distinct roots, if a(2,2).
Hence, option 'B' is correct.

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