Problem:
A mapping(or function) F is said to be contractive if there exists a number lambda less than 1 such that |F(x) - F(y)| <= lambda |x-y|, determine the best value of lambda for f(x) = |x|^(3/2) on |x| <= 1/3 interval.
Using the MVT, |f(x) - f(y)| = f'(z) |x - y|, I've been able to solve similar problems such as the one above by looking for maximums for f'(z). However, I can not take the derivative of |x|^(3/2). So, I just used x^(3/2) and plugged (1/3) into its derivative of (3*sqrt(x))/ 2 to get sqrt(3)/2 for my lambda value. Would this be correct? If not, any ideas as to how I can solve this problem? Thanks.
A mapping(or function) F is said to be contractive if there exists a number lambda less than 1 such that |F(x) - F(y)| <= lambda |x-y|, determine the best value of lambda for f(x) = |x|^(3/2) on |x| <= 1/3 interval.
Using the MVT, |f(x) - f(y)| = f'(z) |x - y|, I've been able to solve similar problems such as the one above by looking for maximums for f'(z). However, I can not take the derivative of |x|^(3/2). So, I just used x^(3/2) and plugged (1/3) into its derivative of (3*sqrt(x))/ 2 to get sqrt(3)/2 for my lambda value. Would this be correct? If not, any ideas as to how I can solve this problem? Thanks.