Question

Which of the following interval(s) satisfy the inequality $x^{20}-x^{11}+x^6-x^3+1>0$

• A. $(-\infty ,-2)$
• B. $(2,\infty )$
• C. $(-\infty ,1)$
• D. $(0,\infty )$

Answer 1

Answer: (A), (B), (C), (D)

The correct options are
A $(-\infty ,-2)$
B $(2,\infty )$
C $(-\infty ,1)$
D $(0,\infty )$

Case (i): $x<0$
$x<0\to \underset{+}{(x^{20}-x^{11})}+\underset{+}{(x^6-x^3)}>0$
$\therefore x<0$ satisfies.
Case (ii): $x>1$
$x>1\to \underset{+}{(x^{20}-x^{11})}+\underset{+}{(x^6-x^3)}>0$
$\therefore x>1$ satisfies.
Case (iii): $0<x<1$
$0<x<1\to \underset{+}{(1-x^3)}+\underset{+}{(x^6-x^{11})}+\underset{+}{x^{20}}>0$
$\therefore 0<x<1$ satisfies.
So the given equation is satisfied for $\forall x\in R$