JEE Advance - Mathematics (1996 - No. 19)
The real numbers $${x_1}$$, $${x_2}$$, $${x_3}$$ satisfying the equation $${x^3} - {x^2} + \beta x + \gamma = 0$$ are in AP. Find the intervals in which $$\beta \,\,and\,\gamma $$ lie.
$$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
$$\beta \, \in \left[ { - \infty ,\,{1 \over 3}} \right),\,\gamma \, \in \,\left( { - {1 \over {27}},\infty } \right]$$
$$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right),\,\gamma \, \in \,\left[ { - {1 \over {27}},\infty } \right)$$
$$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left( { - {1 \over {27}},\infty } \right)$$
$$\beta \, \in \left( { - \infty ,\,{1 \over 3}} \right],\,\gamma \, \in \,\left[ { - {1 \over {27}},{1 \over {27}}} \right]$$