For a function to be an increasing function, the derivative of the function should be greater than 0 for a given domain of x. Consider the first function. y=
ex+e−x
2
y′>0 ∴
ex+e−x
2
>0 ⇒ex>e−x ⇒e2x>1 ⇒2x>ln(1) ⇒x>0 ⇒x∊[0,∞) Hence statement 1 is true. Consider the second function. y=
ex+e−x
2
y′>0 ∴
ex+e−x
2
Now, this is true for all real values of x since the above exponential functions are always greater than 0 for all real x. Therefore there sum will also be greater than zero for real x. exe−x>0∀x∊R ∴x∊(−∞,∞) Hence, statement 2 is also true.