First slide

Introduction to sets

Question

$The largest interval for which x^{12} - x^{9} + x^{4} - x + 1 > 0 is$

Moderate

A

$- 4 < x \leq 0$
B

$0 < x < 1$
C

$- 100 < x < 100$
D

$- \infty < x < \infty$

Solution

$Given expression x^{12} - x^{9} + x^{4} - x + 1 = f (x)$
$for x < 0 put x = - y where y > 0$
then we get
$f (- y) = y^{12} + y^{9} + y^{4} + y + 1 > 0 for y > 0$
$\begin{array}{l} For 0 < x < 1, x^{9} < x^{4} \Rightarrow - x^{9} + x^{4} > 0 \\ Also 1 - x > 0 and x^{12} > 0 \end{array}$
$\begin{array}{l} \Rightarrow x^{12} - x^{9} + x^{4} + 1 - x > 0 \Rightarrow f (x) > 0 \\ For x > 1, f (x) = x (x^{3} - 1) (x^{8} + 1) + 1 > 0 \\ Also x = 0 and x = 1 satisfy the inequality. \\ So f (x) > 0 for - \infty < x < \infty \end{array}$

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