Question

Find the local extremum values of the following functions : 

f(x) = (x – 1) (x – 2)²

Answer 1

f(x) = (x – 1)(x – 2)² 

f’(x) = (x – 2)² + 2(x – 1)(x – 2) 

= (x – 2)(x – 2 + 2x – 2) 

= (x – 2)(3x – 4) 

f’’(x) = (3x – 4) + 3(x – 2) 

For maxima and minima, 

f'(x) = 0 

(x – 2)(3x – 4) = 0 

x = 2, \(\frac{4}{3}\)

Now,

f’’(2) > 0 x = 2 is point of local minima 

f’’(\(\frac{4}{3}\)) = – 2 < 0 

x = \(\frac{4}{3}\)is point of local maxima 

Hence,

local max value = f(\(\frac{4}{3}\)) = \(\frac{4}{27}\)

local min value = f(2) = 0