The least value of the function f(x)=a x+ b x , (x>0, b>0, a>0) is

Step-by-step solution

The function is given by f(x) = ax + b x for x > 0 , a > 0 , and b > 0 . Since ax > 0 and b x > 0 , we can apply the Arithmetic Mean-Geometric Mean (AM-GM) inequality, which states that for positive real numbers u and v , u+v 2 uv . Let u = ax and v = b x . Then, ax + b x 2 ax b x f(x) 2 ab f(x) 2 ab The equality holds when ax = b x , which implies x^2 = b a , or x = b a . Since x > 0 , this value is valid. Thus, the least value of the function is 2 ab . Answer: 2 a b