JEE Mains · Maths · STD 12 - 6. Application of derivatives
Let \(\mathrm{f}(\mathrm{x})\) be a polynomial of degree \(5\) such that \(\mathrm{x}=\pm 1\) are its critical points. \(\mathop {\lim }\limits_{x \to 0} \left(2+\frac{f(x)}{x^{3}}\right)=4,\) then which one of the following is not true?
- A \(f\) is an odd function
- B \(\mathrm{x}=1\) is a point of minima and \(\mathrm{x}=-1\) is a point of maxima of \(\mathrm{f}\).
- C \(\mathrm{x}=1\) is a point of maxima and \(\mathrm{x}=-1\) is a point of minimum of \(\mathrm{f}\).
- D \(f(1)-4 f(-1)=4\)
Answer & Solution
Correct Answer
(B) \(\mathrm{x}=1\) is a point of minima and \(\mathrm{x}=-1\) is a point of maxima of \(\mathrm{f}\).
Step-by-step Solution
Detailed explanation
\(\lim _{x \rightarrow 0}\left(2+\frac{f(x)}{x^{3}}\right)=4\) \(f(x)=2 x^{3}+a x^{4}+b x^{5}\) \(f^{\prime}(x)=6 x^{2}+4 a x^{3}+5 b x^{4}\) \(f^{\prime}(1)=0, f^{\prime}(-1)=0\) \(a=0, b=\frac{-6}{5} \Rightarrow f(x)=2 x^{3}-\frac{6}{5} x^{5}\)…
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