Find the range of values of $^{'} a^{'}$ for which the function $f (x) = x^{3} + (2 a + 3) x^{2} + 3 (2 a + 1) x + 5$ is monotonic in $R$ Hence find the set of values of $^{'} a^{'}$ for which $f (x)$ in invertible.

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Solution

F(x) is monotonic in R
That implies f'(x) is either positive or negative for R (may be zero at a specific point)
$f^{'} (x) = 3 x^{2} + 2 (2 a + 3) x + 3 (2 a + 1) \geq 0$
$\to D \leq 0$
$\to (2 a + 3)^{2} - 9 (2 a + 1) \leq 0$
Solving we get $a \in [0, \frac{3}{2}]$

And since f(x) is a cubic polynomial it is onto function .
Thus it is invertible for $a \in [0, \frac{3}{2}]$

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