Question 5: The function y=f(x) has a graph of the function y=f'(x) as shown in the figure. Ask the graph of the function \(y=f(2x^2+1)\) in which of the following ranges?

From the graph of the function y=f'(x), the function y=f(x) covariates on intervals \((-1;+\infty)\). I guess \(f'(x)>0\Leftrightarrow x>-1\) and \(f'(x)<0\Leftrightarrow x<-1\).

Consider the function \(y=f(2x^2+1)\Rightarrow y’=4x.f'(2x^2+1)\).

Constan \(y=f(2x^2+1)\) covariates if and only if \(y’=4x.f'(2x^2+1)>0\)

\( \Leftrightarrow \left[\begin{array}{l}\left\{\begin{array}{l}x>0\\f’\left({2{x^2}+1}\right))>0\end{array}\right\\\left\{\begin{array}{l}x<0\\f'\left({2{x^2}+1}\right)<0\end{array}\right\end{array}\right\Leftrightarrow\left[\begin{array}{l}\left\{\begin{array}{l}x>0\\2{x^2}+1>-1\end{array}\right\\\left\{\begin{array}{l}x<0\\2{x^2}+1<-1\end{array}\right\end{array}\right\Leftrightarrowx>0\)

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