Answer to Question #200205 in Real Analysis for Rajkumar

Question

Show that there is no real number, k for which the equation, x⁴ - 3x² + k =0 has two distinct roots in the interval [2,3].

Answer 1

Given

x⁴ - 3x² + k =0

we want to find value of k, for which the equation, x⁴ - 3x² + k =0 has two distinct roots in the interval [2,3].

we write equation in this form

$(x^2)^2-3(x^2)+k=0$

let two distinct root is x₁ and x₂

according to question

$2\leq x_1, x_2 \leq 3\\ 4\leq x_1^2, x_2^2 \leq 9\\$

we know ,

$x=\frac{-b\pm\sqrt{b^2-4ac}}{2a} \\$

hence

$4\leq \frac{3\pm\sqrt{9-4k}}{2} \leq 9\\ 8\leq {3\pm\sqrt{9-4k}}{} \leq 18\\ 5\leq \pm\sqrt{9-4k}\leq 15.......(1) \\ now \space take \space + or- \space one by one\\ 5\leq \sqrt{9-4k}\leq 15....(2)\\ and\\ 5\leq -\sqrt{9-4k}\leq 15\\ or\\ -5\geq \sqrt{9-4k}\geq -15.....(3)\\ from(2) and(3)\\ there\space no\space existing\space k \space which \space \\satisfied \space eq(2)and(3) \\$

hence we say that

there is no real number, k for which the equation, x⁴ - 3x² + k =0 has two distinct roots in the interval [2,3].