Question

if 2^(1-x)+2^(1+x),f(x),3^(x)+3^(-x)are in AP then minimum value of f(x) is
OPTIONS:
(A)1
(B)2
(C)3
(D)4​

Answer 1

The minimum value of f(x) is equal to 3.

• First of all to find f(x) we make use of the fact that the second term of an AP is the average of the first and third terms.
• So, f(x)= (2^(1-x) + 2^(1+x) +3^x +3^-x)/2

          ⇒f(x)= (2^-x + 2^x) + (3^x +3^-x)/2

• The function f(x) can be treated as the combination of two functions of the type (a^x + a^-x).
• The function (a^x + a^-x) takes the minimum value of 2 when x=0,otherwise increases on either side of x=0.
• Hence f(x) will also attain its minimum value at x=0, which is equal to 3.