Application of Derivatives 2 Question 8

####8. If $f(x)=x^{3}+b x^{2}+c x+d$ and $0<b^{2}<c$, then in $(-\infty, \infty)$

(a) $f(x)$ is strictly increasing function

(2004, 2M)

(b) $f(x)$ has a local maxima

(c) $f(x)$ is strictly decreasing function

(d) $f(x)$ is bounded

Show Answer

Answer:

(a)

Solution:

  1. Given,

$ f(x)=x^{3}+b x^{2}+c x+d $

$\Rightarrow \quad f^{\prime}(x)=3 x^{2}+2 b x+c$

As we know that, if $a x^{2}+b x+c>0, \forall x$, then $a>0$ and $D<0$.

Now, $\quad D=4 b^{2}-12 c=4\left(b^{2}-c\right)-8 c$

[where, $b^{2}-c<0$ and $c>0$ ]

$ \therefore \quad D=(-\mathrm{ve}) \text { or } D<0 $

$\Rightarrow f^{\prime}(x)=3 x^{2}+2 b x+c>0 \forall x \in(-\infty, \infty)$ [as $D<0$ and $a>0$ ]

Hence, $f(x)$ is strictly increasing function.



NCERT Chapter Video Solution

Dual Pane