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

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4-2.Quadratic Equations and Inequations

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

A

$2$

B

$3$

C

$4$

D

$5$

(IIT-2011)

Solution

$f(x)=x^4-4 x^3+12 x^2+x-1$

Let $f(x)$ has four distinct real roots

$\Rightarrow f^{\prime}(x)=4 x^3-12 x^2+24 x+1$

$f ^{\prime}( x )$ has three distinct real roots

$f ^{\prime \prime}( x )=12 x ^2-24 x +24=12\left( x ^2-2 x +2\right) $

$D =-4 < 0$

$f ^{\prime \prime}$ (x) cannot have 2 real solutions.

So, $f ( x )$ cannot have four real distinct roots

It can have 2 or o real roots.

$f(0)=-1, \quad f(1)=9$

$\Rightarrow$ At least one real solution

So, 2 real distinct solutions.

Standard 11

Mathematics

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