JEE Advanced2011MathematicsApplication of DerivativesActual

The number of distinct real roots of x^4-4 x^3+12 x^2+x-1=0 is

Correct answer

Step-by-step solution

f(x)=x^4-4 x^3+12 x^2+x-1 aligned f^ (x) & =4 x^3-12 x^2+24 x+1 f^ (x) & =12 x^2-24 x+24 & =12 (x^2-2 x+2 ) & =12 (x-1)^2+1 >0, for all x aligned f^ (x) is increasing. Since, f^ (x) is cubic and increasing. f^ (x) has only one real root and two imaginary roots. f(x) cannot have all distinct root. Atmost 2 real roots. Now, f(-1)=15 , f(0)=-1 and f(1)=9 f(x) must have one root in (-1,0) and other in (0,1) . 2 real roots.

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