MHT CET · Maths · Application of Derivatives
The function \(\mathrm{f}(x)=2 x^3-6 x+5\) is an increasing function, if
- A \(0 \lt x \lt 1\)
- B \(-1 \lt x \lt 1\)
- C \(x \lt -1\) or \(x\gt1\)
- D \(-1 \lt x \lt -\frac{1}{2}\)
Answer & Solution
Correct Answer
(C) \(x \lt -1\) or \(x\gt1\)
Step-by-step Solution
Detailed explanation
\(\begin{aligned}
& f(x)=2 x^3-6 x+5 \\
\therefore \quad & f^{\prime}(x)=6 x^2-6
\end{aligned}\)
For \(\mathrm{f}(x)\) to be increasing, \(\mathrm{f}^{\prime}(x)\gt0\)
\(\begin{aligned}
& \Rightarrow 6 x^2-6\gt0 \Rightarrow(x-1)(x+1)\gt0 \\
& \Rightarrow x\gt1 \text { or } x \lt -1
\end{aligned}\)
& f(x)=2 x^3-6 x+5 \\
\therefore \quad & f^{\prime}(x)=6 x^2-6
\end{aligned}\)
For \(\mathrm{f}(x)\) to be increasing, \(\mathrm{f}^{\prime}(x)\gt0\)
\(\begin{aligned}
& \Rightarrow 6 x^2-6\gt0 \Rightarrow(x-1)(x+1)\gt0 \\
& \Rightarrow x\gt1 \text { or } x \lt -1
\end{aligned}\)
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