Question

Let $\alpha ,\beta$ and $\gamma$ be the roots of $f\left(x\right)=0$ , where $f\left(x\right)=x^3+x^2-5x-1$ . Then the value of $\left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]$ is equal to (where $\left[\cdot \right]$ denotes the greatest integer function)

• A. $-1$
• B. $1$
• C. $4$
• D. $-3$

Answer 1

Answer: (D)

$f\left(x\right)=x^3+x^2-5x-1$ $f^{{}^{\prime }}\left(x\right)=3x^2+2x-5=\left(x-1\right)\left(3x+5\right)$

$x=-\frac{5}{3}\to$ point of local maxima,
$x=1\to$ point of local minima,
Increasing function in $\left(-\infty ,-\frac{5}{3}\right)\cup \left(1,\infty \right)$
Decreasing function in $\left(-\frac{5}{3},1\right)$

Also, $f\left(0\right)=-1,f\left(1\right)=-4,f\left(2\right)=1,f\left(-1\right)=4$
$f\left(-2\right)=-8+4+10-1=5$
$f\left(-3\right)=-27+9+15-1=-4$
$\Rightarrow \alpha \in \left(-3,-2\right);\beta \in \left(-1,0\right);\gamma \in \left(1,2\right)$
$\Rightarrow \left[\alpha \right]=-3,\left[\beta \right]=-1,\left[\gamma \right]=1$
$\Rightarrow \left[\alpha \right]+\left[\beta \right]+\left[\gamma \right]=-3$