Question

Find all the maxima and minima of the function f(x) = x(x - 1)², (0 ≤ x ≤ 2). Also determine the area bounded by the curve y = x(x - 1)², the y-axis and the line y = 2.

Answer 1

See Fig.

f(x) = x(x - 1)² 

f′(x) = 1 · (x - 1)² + x · 2(x - 1) = (x - 1) (3x - 1) = 0 

⇒x  = 1/3, 

f′′(x) = 1 · (3x - 1) + (x - 1)³ = 6x - 4 

At x = 1, f′′(1) = 2 > 0 min. value = 0

At x = 1/3, f''(1/3) = -2 < 0 max. value = 4/17