UPSC NDA I 2015 Mathematics Solved Paper

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.