1.

Consider the polynomialf(x)=1 + 2x + 3x^2 +4x^3for all x in R

Answer»

`(-1//4,0)`
`(-11 ,-3 //4)`
`(-3 //4, -1//2)`
`(0 ,1//4)`

Solution :We have
`f(x)=1+2x+3x^2+4x^2`
`rArr f(x)=2+6x+12x^2=2(6x^2+3x+1) gt 0` for all `x in R`
`rArrf(x)` is striclly increasing on R
Now `f(-3//4) lt 0` and `f(-1//2) gt 0`
`rArr ` f(x) has a REAL root between `-3/4 " and" -1/2`
ALSO f(x) is stricly increasing on R. So f(x) has exactly one real root in the interval `(-3/4, -1/2)`


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