For the function [f(x) = 4 _e(x - 1) - 2x^2 + 4x + 5,,x 1], which one of the following i…

Question

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

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