The largest interval for which $x^{22} - x^{19} + x^{14} - x^{5} + 1 > 0$ is

(0,1)

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$(- \infty, 1]$

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$[1, \infty)$

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$(- \infty, \infty)$

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Solution

The correct option is D
$(- \infty, \infty)$

If $x \leq 0$
$x^{22} \geq 0, x^{14} \geq 0, x^{19} \leq 0, x^{5} \leq 0 \Rightarrow x^{22} - x^{19} + x^{14} - x^{5} + 1 > 0$

If $0 < x < 1$ , then $x^{5} < 1$ and hence
$x^{22} - x^{19} + x^{14} - x^{5} + 1 = x^{22} + x^{14} (1 - x^{5}) + (1 - x^{5})$
So, $x^{22} - x^{19} + x^{14} - x^{5} + 1 > 0$ when $0 < x < 1$ .

If $x > 1$ ; then $x^{3} > 1$
$x^{22} - x^{19} + x^{14} - x^{5} + 1 = x^{19} (x^{3} - 1) + x^{5} (x^{9} - 1) + 1 > 0$

Hence, $x^{22} - x^{14} - x^{5} + 1 > 0$ for $- \infty < x < \infty .$

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