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The largest interval for whichx^(12)+x^9...

The largest interval for which`x^(12)+x^9+x^4-x+1>0` `-4

A

`(-4, 0]`

B

`(-100, 100)`

C

`(0, 1)`

D

`(-oo, oo)`

Text Solution

Verified by Experts

The correct Answer is:
D
We have,
`x^(12) - x^(9) + x^(4)-x + 1`
`=x^(9) (x^(3) - 1) +x(x^(3) -1) + 1`
` = x(x^(3) -1)(x^(8) + 1) + 1 ge 1 gt 0 "for" x ge 1.`
Also,
`x^(12) -x^(9) +x^(4) -x +1 =x^(12) + x^(4)(1-x^(5)) + (1-x) gt 0 "for" 0 le x lt 1`
Clearly, `x^(12)-x^(9) + x^(4)-x+1 gt 0 "for all "x lt 0.`
Hence, `x^(12)-x^(9) + x^(4) -x + 1 gt 0"for all" x in (-oo, oo).`
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