MHT CET · Maths · Application of Derivatives
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 \((-\infty, \infty)\)
- A \(\mathrm{f}(x)\) 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
Answer & Solution
Correct Answer
(A) \(\mathrm{f}(x)\) is strictly increasing function
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& f(x)=x^3+b x^2+c x+d \\
& f^{\prime}(x)=3 x^2+2 b x+c
\end{aligned}\)
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 \(R\).
& f(x)=x^3+b x^2+c x+d \\
& f^{\prime}(x)=3 x^2+2 b x+c
\end{aligned}\)
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 \(R\).
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