Find all real values of $m$ for which the inequality $m x^{2} - 4 x + 3 m + 1 > 0$ is satisfied for all positive $x$ .

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Solution

$m x^{2} - 4 x + 3 m + 1 > 0 \forall x ϵ R$
$m > 0$
$b < 0$
$(- 4)^{2} - 4 m (3 m + 1) < 0$
$4 - 3 m^{2} - m < 0$
$3 m^{2} + m - 4 > 0$
$3 m^{2} + 4 m - 3 m - 4 > 0$
$(m - 1) (3 m + 4) > 0$
$m ϵ (- \infty, \frac{- 4}{3}) \cup (1, \infty)$
$m ϵ (1, \infty)$

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