Application of Derivatives 4 Question 63
66. The number of distinct real roots of $x^{4}-4 x^{3}+12 x^{2}+x-1=0$ is……
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Solution:
- $f(x)=x^{4}-4 x^{3}+12 x^{2}+x-1$
$f^{\prime}(x)=4 x^{3}-12 x^{2}+24 x+1$
$f^{\prime \prime}(x)=12 x^{2}-24 x+24=12\left(x^{2}-2 x+2\right)$
$$ =12{(x-1)^{2}+1 }>0 \forall x $$
$\Rightarrow \quad f^{\prime}(x)$ is increasing.
Since, $f^{\prime}(x)$ is cubic and increasing.
$\Rightarrow f^{\prime}(x)$ has only one real root and two imaginary roots.
$\therefore f(x)$ cannot have all distinct roots.
$\Rightarrow$ Atmost 2 real roots.
Now, $f(-1)=15, f(0)=-1, f(1)=9$
$\therefore f(x)$ must have one root in $(-1,0)$ and other in $(0,1)$.
$\Rightarrow 2$ real roots.
