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
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