local and global extrema

Question

Determine all local and global extrema of the function

f(x)=x(x−1)

on [0,1].

Answer 1

The polynomial function is f(x) = x(x-1) = x² - x

Apply first derivative with respect to x.

f '( x ) = 2x - 1

Apply second derivative with respect to x.

f ''( x ) = 2

Find the critical points where f(x) does not exist or Set f '( x ) = 0

f '( x ) = 0  ⇒ 2x - 1 = 0  ⇒ 2x = 1 ⇒ x = ½.

From Extreme value theorem,

       If f " (x) > 0 (positive) ------> minimum point.

       If f " (x) < 0 (negative) ------> maximum point.

f ''( x ) = 2 > 0 (positive) ------> minimum point exists at critical point.

To find the minimum point of f(x) substitute critical value in x = 1/2 in f(x).

At x = 2 ⇒ f(1/2) = (1/2)² - (1/2) = (1/4)-(1/2) = (1-2)/4 = -1/4 [ minimum ]

 

f(x) values at end points [0,1]

At x = 0 ⇒ f(0) = (0)² - (0) = 0. [ maximum ]

At x = 1 ⇒ f(1) = (1)² - (1) = 1-1 = 0. [ maximum ]

Solution :

Minimum point occurs at x = ½

Maximum point occurs at x = 0 and 1.