Question

Consider the cubic $f\left(x\right)=8x^3+4a{x}^2+2bx+a$, where $a,b,\in R$.

For $a=1$, if $y=f\left(x\right)$ is strictly increasing $\forall x\in R$, then maximum range of values of $b$ is

• A. $(-\infty ,\frac{1}{3})$
• B. $(\frac{1}{3},\infty )$
• C. $[\frac{1}{3},\infty )$
• D. $(-\infty ,\infty )$

Answer 1

Answer: (B)

$(\frac{1}{3},\infty )$

$f\left(x\right)=8x^3+4x^2+2bx+1$

$f^{\prime }\left(x\right)=24x^2+8x+2b$

for $f\left(x\right)$ to be strictly increasing

$f^{\prime }\left(x\right)>0,\forall x\in R$

$\Rightarrow 24x^2+8x+2b>0$

$\Rightarrow b>-12x^2-4x=-12(x^2+2\cdot \frac{1}{6}\cdot x+\frac{1}{36}-\frac{1}{36})$

$\Rightarrow b>\frac{1}{3}-12(x-\frac{1}{6}{)}^2$

Hence range of b is $(\frac{1}{3},\infty )$