<s>[INST] Value of 'a' for which $f(x)=\frac{x^3}{3} +\frac{x^2}{2} +ax+b $ is injective Let $f(x)=\frac{x^3}{3} +\frac{x^2}{2} +ax+b   \forall   x \in R$.
Find the least value of a for which $f(x)$ is injective function.
My approach , for $f(x)$ to be injective either f(x) should be increasing or decreasing function.'
$Y=f'(x)=x^2+x+a$ [Increasing function]
If $Y>0$, then $x^2+x+a>0$, it is possible when $1-4a<0$ or $a>\frac{1}{4}$
If $Y<0$, then $x^2+x+a<0$, it is possible when $1-4a \ge 0$ or $a \le \frac{1}{4}$
The options are (A) $\frac{1}{4}$ which is the correct option, the other given options are (B) $1$, (C) $\frac{1}{2}$, (D) $\frac{1}{8}$
Though my optin is correct but i have one doubt when the equality=$0$, then Y=$0$ hence it is neither increasing nor decreasing function.
 [/INST] Your idea of computing $f'$ is fine, but your computation is wrong. In fact$$f'(x)=x^2+x+a=\left(x+\frac12\right)^2+a-\frac14.$$Can you take it from here?
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