Let A and G denote the arithmetic mean and geometric mean of positive real numbers 5^x and 5^ 1-x . Then the minimum value of the expression 5^x+5^ 1-x where x R is

Step-by-step solution

Let a = 5^x and b = 5^ 1-x . Since x is a real number, a and b are positive real numbers. The arithmetic mean A is given by A = 5^x + 5^ 1-x 2 and the geometric mean G is given by G = 5^x 5^ 1-x . Calculating G : G = 5^ x + 1 - x = 5^1 = 5 . By the AM-GM inequality, for any two positive real numbers a and b , a+b 2 ab . Substituting the values: 5^x + 5^ 1-x 2 5 . Multiplying by 2: 5^x + 5^ 1-x 2 5 . The minimum value of the expression 5^x + 5^ 1-x is 2 5 , which occurs when 5^x = 5^ 1-x , i.e., x = 1-x , or x = 0.5