Find the local extremum values of the following functions : f(x) =x√(1-x) ,x ≤ 1 ​

Question

Find the local extremum values of the following functions :

f(x) = x\(\sqrt{1-x}\), x ≤ 1

Answer 1

f(x) = x\(\sqrt{1-x}\) 

For maxima and minima, 

f'(x) = 0

\(\frac{2-3x}{2\sqrt{1-x}}\) = 0

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

Now,

f’’(\(\frac{2}{3}\)) < 0 

x = \(\frac{2}{3}\) is point of maxima 

Hence,

local max value = f(\(\frac{2}{3}\)) = \(\frac{2}{3\sqrt3}\)