Question

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
• B. $ f(x) $ has a local maxima
• C. $ f(x) $ strictly decreasing function
• D. $ f(x) $ is bounded

Answer 1

Answer: (A)

Given, $f(x)=x^{3}+b x^{2}+c x+d$
$\Rightarrow f'(x)=3 x^{2}+2 b x+c$
(As we know, if $a x^{2}+b x+c>0$ for all $x$
$\Rightarrow a > 0$ and $D<0$ )
Now, $D=4 b^{2}-12 c=4\left(b^{2}-c\right)-8 c$
(where $b^{2}-c < 0$ and $c > 0)$
$\therefore D=(-v e) i e, D < 0$
$\Rightarrow f'(x)=3 x^{2}+2 b x+c>0$
for all $x \in(-\infty, \infty).$
(as $D < 0$ and $a > 0$ )
Hence, $f(x)$ is strictly increasing function.