Question

If function $f\left(x\right)=x^3+\lambda x^2-\lambda x+1$ is increasing at $x=0$ & decreasing at $x=1$, then find the greatest integer value of $\lambda$.

• A. $-4$
• B. $2$
• C. $0$
• D. $-1$

Answer 1

The correct option is A $-4$

$f\left(x\right)=x^3+\lambda x^2-\lambda x+1$
$\Rightarrow f^{\prime }\left(x\right)=3x^2+2\lambda x-\lambda$
Now for f to be increasing at $x=0$ and decreasing at $x=1$
we must have,
$f^{\prime }\left(0\right)>0$ and $f^{\prime }\left(1\right)<0$
$\Rightarrow -\lambda >0$ and $3+2\lambda -\lambda <0$
$\Rightarrow \lambda <0$ and $\lambda <-3$
Hence $\lambda <-3$
Thus greatest integral value of $\lambda$ is $-4$