Question

Find the numbers of integers satisfying the following Inequalities
$\left(i\right)x^4-5x^2+4\le 0$
$\left(ii\right)x^4-2x^2+63\le 0$

Answer 1

$\left(i\right)x^4-5x^2+4\le 0$

$\Rightarrow x^4-4x^2-x^2+4\le 0$

$\Rightarrow x^2(x^2-4)-1(x^2-4)\le 0$

$\Rightarrow (x^2-4)(x^2-1)\le 0$

$\Rightarrow (x-2)(x+2)(x-1)(x+1)\le 0$

The critical points are $\{-2,-1,1,2\}\in Z$

Hence numbers of integers satisfying the inequalities$=4$

$\left(ii\right)x^4-2x^2+63\le 0$

$\Rightarrow x^4-2x^2+1-1+63\le 0$

$\Rightarrow {(x^2-1)}^2+62\le 0$

$\Rightarrow {(x^2-1)}^2+\le -62$

$\Rightarrow x^2-1\le \pm \sqrt{-62}$

Thus, there exists no integer.Hence numbers of integers satisfying the inequalities$=0$