Consider that `f : R rarr R`
(i) Let `f(x) = x^(3) + x^(2) + ax + 4` be bijective, then find a.
(ii) Let `f(x) = x^(3) + bx^(2) + cx + d` is bijective, then find the condition.

Explanation

(i) Clearly f(x) is onto as range `= (-oo, oo)` = co-domain, but for one-one `f(x) = 3x^(2) + 2x + a ge 0`

(ii) f(x) is clearly onto as its range is `(-oo, oo)` = co domain.
But for one-one f'(x) `= 3ax^(2) + 2bx c ge 0 AA, x in R` or `3ax^(2) + 2bx + c le 0` to be so.
`rArr 4b^(2) - 4 xx 3a xx c le 0`
`rArr b^(2) - 3ac le 0`
To prove f to be bijective f should be one-one and onto both.