Question

If 2^(1-x) +2^(1+x) ,f(x),3^x+3^(-x) are in A.P ,then the minimum value of f(x) is ?????

Answer 1

Answer:

3

Step-by-step explanation:

given:

f(x)= (2^(1-x) + 2^(1+x) + 3^x+3^(-x)) /2

f(x)= (2^(1-x) + 2^(1+x))/2 + (3^x+3^(-x)) /2

f(x)= (2^(1-x-1) + 2^(1+x-1)) + (3^x+3^(-x)) /2

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

Now we know that,

A.M≥G.M

where, A.M=arithmetic mean G.M=geometric mean

let us suppose A be a +ve real number.thus, 1/ A would be it's reciprocal which indeed which will also be a +ve real number.

then,by A.M≥G.M inequality,

(A+1/A)/2 ≥ √(A.1/A)

or, A+1/A ≥2

now,we know that a^x > 0 if a>0.

now,we know that a^x > 0 if a>0.so,2^x and 2^(-x), both are +ve real number as 2>0.

=> (2^x + 2^(-x)) ≥ 2

similarly,3^x and 3^(-x) ,both are +ve real number as 3>0.

=> (3^x+3^(-x)) ≥2

or, (3^x+3^(-x)) /2 ≥ 2/2 (dividing both sides by 2)

or, (3^x+3^(-x)) /2 ≥1

thus,f(x) would be minimum when (2^x + 2^(-x)) and (3^x+3^(-x)) /2 are minimum.

=>the minimum value of f(x) will be 3.

Some additional information:

*A.M-G.M inequality can be used only when radicand is positive.

*AM-GM inequality has great importance in senior secondary mathematics and is also helpful in solving olympiads questions.