The critical point(s) of any function #g# occur at #x=c# when #g'(c)=0#. Let's consider a critical point #x=a# for the function #f#, that is #f'(c)=0#. The coordinates of this critical point is given by #(c,f'(c))#.
Consider the function #g(x)=af(x)#. Its critical point occurs when #g'(x)=0#, that is #af'(x)=0#. Either #a=0#, which is false, or #f'(x)=0#, which is true at least for some point #x=c# i.e. #f'(c)=0#. The coordinates of this critical point therefore is given by #(c,g(c))#, or in terms of #f#, #(c, af(x))#.
Consider the function #g(x)=a+f(x)#. Its critical point occurs when #g'(x)=0#, that is #0+f'(x)=0#, which is true at least for some point #x=c# i.e. #f'(c)=0#. The coordinates of this critical point therefore is given by #(c,g(c))#, or in terms of #f#, #(c, a+f(x))#.
Consider the function #g(x)=f(ax)#. Its critical point occurs when #g'(x)=0#, that is #af'(ax)=0#. Either #a=0#, which is false, or #f'(ax)=0#. We know that #f'(c)=0#, so #ax=c# for some #c#, and #x=\frac{c}{a}#. That is, #g'(\frac{c}{a})=0# The coordinates of this critical point therefore is given by #((\frac{c}{a},g((\frac{c}{a}))#, or in terms of #f#, #((\frac{c}{a}, f(c))#.
Consider the function #g(x)=f(a+x)#. Its critical point occurs when #g'(x)=0#, that is #f'(a+x)=0#, which is true at least for some point #a+x=c# i.e. #f'(c)=0#. Thus, #x=c-a# and #g'(c-a)=0#. The coordinates of this critical point therefore is given by #(c-a,g(c-a))#, or in terms of #f#, #(c-a, f(c))#.
