Question

The function $f(x)=x^3+6x^2+(9+2k)x+1$ is strictly increasing for all $x$, if

• A. $k>\frac{3}{2}$
• B. $k\ge \frac{3}{2}$
• C. $k<\frac{3}{2}$
• D. $k\le \frac{3}{2}$

Answer 1

Answer: (A)

Here, $f\left(x\right)=x^3+6x^2+\left(9+2k\right)x+1$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+12x+9+2k$.
Now $f$ is strictly increasing for all $x\in R$
if $f^{\prime }\left(x\right)>0\forall x\in R$
i.e. if $3x^2+12x+\left(9+2k\right)>0$ for all $x\in R$
i.e. if $12^2-4\cdot 3\left(9+2k\right)<0$
i.e. if $-24k<-36$ i.e. if $k>\frac{3}{2}$.