# The graph of the first derivative

The following graph shows the first derivative of $f(x)$, i.e., $f'(x)$. The function $f$ itself is defined in $[-5,5]$. A function $g$ is defined as: $g(x)=f(x)-x$. How many minimum and maximum points does $g$ has? I don't even know where to start. Your help will be most appreciated!

• What is the connection between the extrema of a function and its derivative ? – Yves Daoust Jan 30 '17 at 19:47
• How is the derivative of $g$ related to the derivative of $f$? – John Hughes Jan 30 '17 at 19:47
• Oh, I see what you mean. g'(x)=f'(x)+1, then I need f'(x)=-1. And there is only one such a point ? Am I correct? – user2899944 Jan 30 '17 at 20:21

Hint: $g'(x)=f'(x)-1$. You are interested in what is going on around points where $g'(x)=0$, i.e., $f'(x)=1$. Use the fact that for a local extremum to occur, the sign of the first derivative must change around the point where the derivative is zero.