Q73 / 150 · MHT CET 2024 (03 May Shift 2) · Mathematics
If $\mathrm{f}(x)=x^3+\mathrm{b} x^2+\mathrm{c} x+\mathrm{d}$ and $0 \lt \mathrm{b}^2 \lt \mathrm{c}$, then in $(-\infty, \infty)$
- Correct answerA$\mathrm{f}(x)$ is strictly increasing function
- B$\mathrm{f}(x)$ is bounded
- C$\mathrm{f}(x)$ has a local maxima
- D$\mathrm{f}(x)$ is a strictly decreasing function
Solution
\begin{aligned}
& f(x)=x^3+b x^2+c x+d \\
& f^{\prime}(x)=3 x^2+2 b x+c
\end{aligned}
$\therefore$
Now its discriminant $=4\left(b^2-3 c\right)$ $\Rightarrow 4\left(b^2-\mathrm{c}\right)-8 \mathrm{c} \lt 0$, as $\mathrm{b}^2 \lt \mathrm{c}$ and $\mathrm{c}\gt0$ $\Rightarrow \mathrm{f}^{\prime}(x)\gt0$ for all $x \in \mathrm{R}$ $\Rightarrow f$ is strictly increasing on $R$.
Now its discriminant $=4\left(b^2-3 c\right)$ $\Rightarrow 4\left(b^2-\mathrm{c}\right)-8 \mathrm{c} \lt 0$, as $\mathrm{b}^2 \lt \mathrm{c}$ and $\mathrm{c}\gt0$ $\Rightarrow \mathrm{f}^{\prime}(x)\gt0$ for all $x \in \mathrm{R}$ $\Rightarrow f$ is strictly increasing on $R$.
Hence, the correct answer is option: 'A'.