#y = (x-1)^4#

Power rule + chain rule

#y' = 4(x-1)^3 * 1#

#y' = 4(x-1)^3#

Critical points are when #f'(x) = 0# so...

we set #y' = 0#

#0 = 4(x-1)^3#

#0/color (red) 4 = (4(x-1)^3)/color(red)4#

#0 = (x-1)^3#

#0^(color(red)(1/3)=((x-1)^3)^color(red)(1/3)#

#0color(red)+color(red)1 = x - 1 color(red) +color(red)1#

#1 = x#

We now know that #x=1# is a critical point.

Now we look at the original function #y = (x-1)^4#

We can take a look at #x=1# and compare that to #x# slightly smaller than 1 and #x# slightly larger than 1

#y = (1-1)^4#

#y = (0)^4#

#y = 0#

#y = (0.99-1)^4#

#y = (-0.01)^4#

#y = 0.00000001#

#y = (1.01-1)^4#

#y = (.01)^4#

#y = 0.00000001#

The points immediately to the left and right of #x=1# are both larger than #x=1# therefore #x=1# must be a minimum