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Q.
The function $f(x) = x^3 + 6x^2 + (9 + 2 \,k) x + 1$ is strictly increasing for all $x$, if
Application of Derivatives
Solution:
Here, $f\left(x\right) = x^{3}+6x^{2}+\left(9+2k\right)x+1$
$\Rightarrow f'\left(x\right) = 3x^{2}+12\, x + 9 +2k$.
Now $f$ is strictly increasing for all $x \in R$
if $f'\left(x\right) > 0\,\forall \,x \in R$
i.e. if $3x^{2 }+ 12 \,x + \left(9 + 2k\right) > 0$ for all $x \in R$
i.e. if $12^{2} - 4\cdot3 \left(9 + 2k\right) < 0 $
i.e. if $-24\, k < -36$ i.e. if $k > \frac{3}{2}$.