Question

Let $f: R \rightarrow R$ be defined as $f(x)=\frac{x^3}{3}+\frac{x^2}{2}+a x+b$. The least value of ' $a$ ' for which $f(x)$ is injective function, is

• A. $ \frac{1}{4}$
• B. $\frac{1}{8}$
• C. $\frac{1}{2}$
• D. 1

Answer 1

Answer: (A)

If $f ( x )$ is one-one then $f ( x )$ must be monotonic.
Now, $f^{\prime}(x)=x^2+x+a \geq 0 \forall x \in R$
$\Rightarrow D \leq 0$ i.c. $1-4 a \leq 0 \Rightarrow a \geq \frac{1}{4}$
So, $a_{\min }=\frac{1}{4}$.