$f (x) = \frac{x^{3}}{3} + \frac{x^{2}}{2} + a x + b, \forall x ϵ R$ , then find least value of ‘a’ for which f(x) is injective function

\frac{1}{4}

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\frac{1}{2}

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\frac{1}{8}

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Solution

The correct option is A $\frac{1}{4}$
$f^{^{'}} (x) = x^{2} + x + a$
$f^{^{'}} (x) \geq 0$
$\Rightarrow {(x + \frac{1}{2})}^{2} + (a - \frac{1}{4}) \geq 0$
hence $a \geq \frac{1}{4}$

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2) $f : R \to R$ defind by $f (x) = [x]$ is	b) surjection but not injection
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