Mathematics

# Find the integral part of the greatest root of equation $x^{3}-10x^{2}-11x-100=0$.

##### SOLUTION
Given equation is

$x^{3}-10 x^{2}-11 x-100=0$

Let $f(x)=x^{3}-10 x^{2}-11 x-100$

$\Rightarrow f^{\prime}(x)=3 x^{2}-20 x-11$

For $3 x^{2}-20 x-11=0,$

we have$x=\dfrac{20\pm\sqrt{400+132}}{6}=\dfrac{10\pm\sqrt{133}}{3}$

Hence, graph of $y=f(x)$ is plotted

Now, $(10+\sqrt{133}) / 3 \cong 7.16$

$f(8)=8^{3}-10(8)^{2}-11(8)-100<0$

$f(9)=9^{3}-10(9)^{2}-11(9)-100<0$

$f(10)=10^{3}-10(10)^{2}-11(10)-100<0$

$f(11)=11^{3}-10(11)^{2}-11(11)-100<0$

$f(12)=12^{3}-10(12)^{2}-11(12)-100>0$

$\Rightarrow \quad y \in(11,12)$

$\Rightarrow \quad[y] \quad=11$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 86

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