Question

The number of real solutions of the equation

cos(e^x) = 2^x + 2^-x is

(a) 0 (b) 1 (c) 2 (d) infinitely many

Answer 1

we all know exponential function is always positive
here , 2⁻ˣ and 2ˣ both are exponential function. so, both are positive.
we also know, all positive term follow
AM≥ GM
so,
(2⁻ˣ + 2ˣ)/2 ≥√(2⁻ˣ.2ˣ)


(2⁻ˣ + 2ˣ) ≥ 2

hence cos(eˣ) = (2⁻ˣ + 2ˣ) ≥2

but we know maximum value of cos function is 1
hence cos(eˣ) ≠ 2

hence no any solution is possible here.
no of solution = 0