\
"Yes, you are exactly right: it can be correctly thought of as"
"a product rule and chain rule problem."
"Let's compute" \ y' ":"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad y \ = \ x (1 - x )^4.
\qquad \qquad \qquad \qquad \quad :. \qquad \qquad \qquad \quad y \ = \ [ x ] \cdot [ (1 - x )^4 ].
"Product Rule:" \qquad \qquad \quad \ y' \ = \ x [ (1 - x )^4 ]' + [ x ]' [ (1 - x )^4 ].
"Chain Rule for first differentiated quantity:"
\qquad \qquad \quad \quad \ y' \ = \ x [ 4 (1 - x )^3 (1 - x )' ] + 1 \cdot [ (1 - x )^4 ].
"Continuing (notice the -1 that appears -- important):"
\qquad \qquad \quad \quad \ y' \ = \ x [ 4 (1 - x )^3 ( -1 ) ] + (1 - x )^4.
"Simplifying -- factor out lowest powers of same quantities:"
\qquad \qquad \quad \quad \ y' \ = \ (1 - x )^3 [x ( 4 ) ( -1 ) + (1 - x )^1 ]
\qquad \qquad \qquad \qquad \ \ = \ (1 - x )^3 [x ( -4 ) + (1 - x ) ]
\qquad \qquad \qquad \qquad \ \ = \ (1 - x )^3 [ -4 x + 1 - x ]
\qquad \qquad \qquad \qquad \ \ = \ (1 - x )^3 ( 1 - 5 x ).
\qquad \qquad \qquad \qquad \ \ = \ ( 1 - 5 x ) (1 - x )^3.
"Thus:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \ \ y' \ = \ ( 1 - 5 x ) (1 - x )^3.
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"Now we can calculate the critical points."
"Recall that the critical points are the points where the derivative"
"vanishes or is undefined."
"So we solve:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad y' = 0 \quad \quad "and" \quad \quad y' = "undefined"
"So we solve:"
\quad ( 1 - 5 x ) (1 - x )^3 = 0 \qquad "and" \qquad ( 1 - 5 x ) (1 - x )^3 = "undefined"
"In the second part of the previous, we note that"
( 1 - 5 x ) (1 - x )^3 \ \ "is defined everywhere, so the second part"
"yields no solutions. So we continue solving only the first part:"
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad ( 1 - 5 x ) (1 - x )^3 = 0
\qquad \qquad \qquad \qquad \qquad \qquad 1 - 5 x = 0 \qquad \qquad (1 - x )^3 = 0
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \quad \ 5 x = 1 \qquad \qquad 1 - x = 0
\qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad x = 1/5 \qquad \qquad x = 1
"These are our critical points."
\
"Summarizing:"
"The critical points of" \ \ y = x (1 - x )^4 \ \ "are:" \qquad x = 1/5, \ 1.