(a) Take the derivative of f(x)
f'(x)=3e^3x - e^-x > 0
3e^3x > e^-x
e^-x (3e^4x-1) >0
e^4x > 1/3
4x> -ln3
x> -1/4 ln 3
(-1/4 ln 3, Infinity) f is increasing
(negative infinity, -1/4 ln 3) f is monotonically decreasing.
(b) f(-1/4 ln 3)= 3^(-3/4)+3^(1/4) = 4* 3^(-3/4)
(c) take the second order derivative of f(x)
f''(x)= 9e^3x + e^-x > 0 for all x
thus the interval is all real numbers R, namely (negative infinity, infinity)