Find the local extremum values of the following functions : f(x) = (x

1.	Find the local extremum values of the following functions : f(x) = (x – 1) (x – 2)2
Answer» 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

Find the local extremum values of the following functions : f(x) = (x – 1) (x – 2)2

Answer»

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