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
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Solution:
- 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 $c>0$ ]
$$ \therefore \quad D=(-ve) \text { or } D<0 $$
$\Rightarrow f^{\prime}(x)=3 x^{2}+2 b x+c>0 \forall x \in(-\infty, \infty)$ [as $D<0$ and $a>0$ ]
Hence, $f(x)$ is strictly increasing function.
