Question 36: Given a function y=f(x) with a graph y=f'(x) as shown in the following figure Function graph (g(x)=left|2 f(x)-x^{2 }right|) how many extremes are there?

Consider the function \(h(x)=2 f(x)-x^{2} \Rightarrow h^{\prime}(x)=2 f^{\prime}(x)-2 x\)

From the graph we see \(h^{\prime}(x)=0 \Leftrightarrow f^{\prime}(x)=x \Leftrightarrow x=-2 \vee x=2 \vee x=4\)

\(\begin{array}{l} \int\limits_{-2}^{2}\left(2 f^{\prime}(x)-2 x\right) dx>\int\limits_{2} ^{4}\left(2 x-2 f^{\prime}(x)\right) dx>0 \\ \left.\Leftrightarrow h(x)\right|_{-2} ^{2}> -\left.h(x)\right|_{2} ^{4} \Leftrightarrow h(2)-h(-2)>-(h(4)-h(2)) \Leftrightarrow h(4) >h(-2) \end{array}\)

Variation table

So \(g(x)=\left|2 f(x)-x^{2}\right|\) have up to 7 extremes