QJEEM22Maadca: JEE Main 2022 Mathematics - Class 12 JEEM22Maadca Solution: For the function , which one of the following is NOT correct?

Question

Verified Solution • 2025–26 Session moderate

For the function

$f(x) = 4{\log _e}(x - 1) - 2{x^2} + 4x + 5,\,x > 1$ , which one of the following is NOT correct?

Accepted Answer

Lets draw the curve

y=f(x)=4 \log _e(x-1)-2 x^2

+4 x+5, x&gt;1

\begin{aligned}&amp;f(x)=4 \log _e(x-1)-2 x^2+4 x+5, x &gt; 1 \\\\&amp;f^{\prime}(x)=\frac{4}{x-1}-4(x-1) \\\\&amp;f^{\prime \prime}(x)=\frac{-4}{(x-1)^2}-4\\\\&amp;\text {For} 1 &lt; x&lt; 2 \Rightarrow f^{\prime}(x) &gt; 0 \\\\&amp;\text {For}~ x &gt; 2 \Rightarrow f^{\prime}(x)&lt;0 \text { (option } A \text { is correct) } \\\\&amp;f(x)=-1 \text { has two solution (option } B \text { is correct) } \\\\&amp;f(e) &gt; 0 \\\\&amp;f(e+1) &lt; 0 \\\\&amp;f(e) \cdot f(e+1)&lt;0(\text { option } D \text { is correct) } \\\\&amp;f^{\prime}(e)-f^{\prime \prime}(2)=\frac{4}{e-1}-4(e-1)+8 &gt; 0\end{aligned}

(option C is incorrect)

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