Question

The value of $\lambda$ for which the function $f\left(x\right)=\lambda x^3-2\lambda x^2+\left(\lambda +1\right)x+3\lambda$ is increasing through out number line.

• A. $\lambda \in \left(-3,3\right)$
• B. $\lambda \in \left(-3,0\right)$
• C. $\lambda \in \left(0,3\right)$
• D. $\lambda \in \left(-1,3\right)$

Answer 1

Answer: (C)

$\lambda \in \left(0,3\right)$

Given,

$f\left(x\right)$$=$ $\lambda x^3-2\lambda x^2+(\lambda +1)x+3\lambda$

for increasing $f^{\prime }\left(x\right)>0$

and $f^{\prime }\left(x\right)=$$3\lambda x^2-4\lambda x+(\lambda +1)$

$3\lambda x^2-4\lambda x+(\lambda +1)$$>0$

for this $D<0$

${(-4\lambda )}^2-4\left(3\lambda \right)(\lambda +1)$$<0$

${16\lambda }^2-(12{\lambda }^2+12\lambda )$$<0$

${16\lambda }^2-12{\lambda }^2-12\lambda$$<0$

$4{\lambda }^2-12\lambda$$<0$

${\lambda }^2-3\lambda$$<0$

$\lambda (\lambda -3)$$<0$

Hence, $\lambda$ $ϵ(0,3)$