If x 2 + m x + 1 x 2 + x + 1 < 3 for all real x , then

Step-by-step solution

We have x 2 + x + 1 = x + 1 2 2 + 3 4 &#62; 0 So, - 3 &#60; x 2 + m x + 1 x 2 + x + 1 &#60; 3 &#8658; - 3 x 2 + x + 1 &#60; x 2 + m x + 1 &#60; 3 x 2 + x + 1 &#8658; 4 x 2 + m + 3 x + 4 &#62; 0 and 2 x 2 + 3 - m x + 2 &#62; 0 for real x &#8658; m + 3 2 - 4 &#215; 4 &#215; 4 &#60; 0 and 3 - m 2 - 4 &#215; 2 &#215; 2 &#60; 0 &#8658; m + 3 + 8 m + 3 - 8 &#60; 0 and m - 3 + 4 m - 3 - 4 &#60; 0 &#8658; m + 11 m - 5 &#60; 0 and m + 1 m - 7 &#60; 0 &#8658; - 11 &#60; m &#60; 5 and - 1 &#60; m &#60; 7 Hence, - 1 &#60; m &#