The number of values of , where the function attains its maximum, is
- A.0
- B.1
- C.2
- D.infinite
Show answer and solution
Correct option: B
Solution
f(x) = cos x + cos(√2 x) attains its maximum value of 2 only when both cosines equal 1 simultaneously.
cos x = 1 requires x = 2πm (integer m), and cos(√2 x) = 1 requires x = 2πn/√2 = π√2 · n (integer n).
So we need 2πm = π√2 · n, giving m/n = √2/2 = 1/√2, which is irrational. Since m and n must be integers, no such pair (m,n) exists — except x = 0, where both conditions give m = 0 and n = 0 simultaneously.
Therefore x = 0 is the one and only point where f(x) reaches its maximum of 2, giving exactly 1 value.
cos x = 1 requires x = 2πm (integer m), and cos(√2 x) = 1 requires x = 2πn/√2 = π√2 · n (integer n).
So we need 2πm = π√2 · n, giving m/n = √2/2 = 1/√2, which is irrational. Since m and n must be integers, no such pair (m,n) exists — except x = 0, where both conditions give m = 0 and n = 0 simultaneously.
Therefore x = 0 is the one and only point where f(x) reaches its maximum of 2, giving exactly 1 value.
[1998]