IIT-JEE 2011 · Paper 2 · Mathematics Q54

MathematicsInteger+4Question 54 of 60

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

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Answer and worked solution

Official answer: 2

Convexity. $f''(x)=12x^2-24x+24=12\left[(x-1)^2+1\right]>0$ , so $f$ is convex and has at most 2 real roots.

Sign change. $f(0)=-1<0$ while $f\to+\infty$ on both sides, so there is exactly one root on each side of 0.

Answer: 2

IIT-JEE 2011 · Paper 2, Mathematics question 54