Decide the monotony of function f(x)=x^27+x^25+e^(x^3)?

The monotony of a function establishes the increasing or
decreasing behaviour of the function.

In order to prove
that f(x) is an increasing function, we have to do the first derivative
test.

If the first derivative of the function is positive,
then the function is increasing.

Let's calculate
f'(x):

 f'(x) =
(x^27+x^25+e^(x^3))'

 f'(x) = (x^27)' + (x^25)' +
(e^(x^3))'

  f'(x) = 27x^26 + 25x^24 +
e^(x^3)*(x^3)'

  f'(x) = 27x^26 + 25x^24 + 2e^(x^3)*(x^2)
> 0

Since each term of the expression
of f'(x) is positive, the sum of positive terms is also a positive expression.
The expression of f'(x) it's obviously>0, so f(x) is an increasing
function