IIT JEE 1982 Maths - MCQ Question 13

${ x }^{ 12 }-{ x }^{ 9 }+{ x }^{ 4 }-x+1$

$\quad =>\quad x({ x }^{ 8 }+1)({ x }^{ 3 }-1)+1$

Here $\quad ({ x }^{ 8 }+1)$ is always a positive number

$\quad ({ x }^{ 3 }-1)$ is +ve for x>1

When x<0 x $\quad ({ x }^{ 3 }-1)$ is greater than zero

Thus ${ x }^{ 12 }-{ x }^{ 9 }+{ x }^{ 4 }-x+1$ > 0 can be easily established for x<0 and x>1

Now for 0<x<1

$\quad x({ x }^{ 8 }+1)({ x }^{ 3 }-1)$ is surely -ve

but from ${ x }^{ 12 }-{ x }^{ 9 }+{ x }^{ 4 }-x+1$

For 0<x<1 x^4 > x^9 => -x^9 + x^4 >0

Further 1 - x > 0 and x^12 is greater than zero

Hence

${ x }^{ 12 }-{ x }^{ 9 }+{ x }^{ 4 }-x+1$ is greater than zero for all real numbers