JEE Mains · Maths · STD 12 - 6. Application of derivatives
\(f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1\), which one of the following is NOT correct?
- A \(f\) is increasing in \((1,2)\) and decreasing in \((2, \infty)\)
- B \(f(x)=-1\) has exactly two solutions
- C \(f'(e) -f^{\prime \prime}(2)<0\)
- D \(f ( x )=0\) has a root in the interval \(( e , e +1)\)
Answer & Solution
Correct Answer
(C) \(f'(e) -f^{\prime \prime}(2)<0\)
Step-by-step Solution
Detailed explanation
\(f(x)=4 \log _{e}(x-1)-2 x^{2}+4 x+5, x>1\) \(f^{\prime}(x)=\frac{4}{x-1}-4(x-1)\) For \(1 < x < 2 \Rightarrow f^{\prime}(x) > 0\) For \(x >2 \Rightarrow f ^{\prime}( x )<0\) (option \(1\) is correct) \(f ( x )=-1\) has two solution (option \(2\) is correct) \(f ( e )>0\)…
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