JEE Mains · Maths · STD 12 - 6. Application of derivatives
If the function \(f:(-\infty,-1] \rightarrow(a, b]\) defined by \(f(x)=e^{x^3-3 x+1}\) is one-one and onto, then the distance of the point \(\mathrm{P}(2 \mathrm{~b}+4, \mathrm{a}+2)\) from the line \(x+e^{-3} y=4\) is :
- A \(2 \sqrt{1+\mathrm{e}^6}\)
- B \(4 \sqrt{1+\mathrm{e}^6}\)
- C \(3 \sqrt{1+\mathrm{e}^6}\)
- D \(\sqrt{1+\mathrm{e}^6}\)
Answer & Solution
Correct Answer
(A) \(2 \sqrt{1+\mathrm{e}^6}\)
Step-by-step Solution
Detailed explanation
\(f(x)=e^{x^3-3 x+1}\) \(f^{\prime}(x)=e^{x^3-3 x+1} \cdot\left(3 x^2-3\right)\) \(=e^{x^3-3 x+1} \cdot 3(x-1)(x+1)\) For \(\mathrm{f}^{\prime}(\mathrm{x}) \geq 0\) \(\therefore \mathrm{f}(\mathrm{x})\) is increasing function \(\therefore a=e^{-\infty}=0=f(-\infty)\)…
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