State whether the function attains a maximum or minimum value (or both) on the interval

Question

State whether the function  f(x)=x^4/4-x^3+x^2 attains a maximum or minimum value (or both) on the interval [0,2). Show your work.

a. max at 1,     min at 0,   and min at 2

b. max at 1,     min at 0

c. min at 1,     max at 0,    and max at 2

d. min at 0,     min at 2,    no max

e. none of the above

Answer 1

f(x)=x^2(x^2/4-x+1)=x^2(x/2-1)^2, so f(x)=0 when x=0 and 2, and f(x)>0 for all x.

There is a maximum between the two zeroes at x=1 which is the axis of symmetry.

So we have minima at (0,0) and (2,0) and a maximum at (1,1/4), answer a.