Question

Set of values for $p$ for which the function given by $f\left(x\right)=x^3-2x^2-px+1$ is one-one function $\forall x\in R$ is

• A. $R$
• B. $(\frac{-4}{3},\infty )$
• C. $(\frac{-4}{3},\frac{4}{3})$
• D. $(-\infty ,\frac{-4}{3})$

Answer 1

The correct option is D $(-\infty ,\frac{-4}{3})$

Given function $f\left(x\right)=x^3-2x^2-px+1$

Now, $f^{\prime }\left(x\right)=3x^2-4x-p$
As the leading co-efficient of $f^{\prime }\left(x\right)$ is positive, so the given function to be one-one, $f^{\prime }\left(x\right)$ has to be greater than $0.$
$f^{\prime }\left(x\right)=3x^2-4x-p>0$
$\Rightarrow 16+12p<0(\because B^2-4AC<0)$
$\Rightarrow p<-\frac{4}{3}$