The largest interval for which`x^(12)+x^9+x^4-x+1>0“-4A. `-4ltx le0`
B. `0ltx lt1`
C. `-100 lt x lt100`
D. `-ooltxltoo`
B. `0ltx lt1`
C. `-100 lt x lt100`
D. `-ooltxltoo`
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Correct Answer – D
Given, `x^(12)-x^(9)+x^(4)-x+1gt0`
here, three cases arises:
Case I When `x le0impliesx^(12)gt0,-x^(9)gt0,x^(4)gt0,-xgt0`
`thereforex^(12)-x^(9)+x^(4)-x+1 gt 0,AA x le0″ “…(i)`
Case II When `0 lt x le 1`
`x^(9)lt x^(4)and x lt1implies-x^(9)+x^(4)gt0and a-xgt0`
`thereforex^(12)-x^(9)+x^(4)-x+1gt0 ltx le1`
Case III When `x gt 1 implies x^(12)gtx^(9)and x^(4)gtx`
`thereforex^(12)-x^(9)+x^(4)-x+1gt0,AAxgt1″ “…(iii)`
From Eqs. (i), (ii), (iii), the above equation holds for all ` x in R.`