Prove that f(x) is increasing for f(x)= e^x + x^3 - x^2 + x

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) = (e^x + x^3 - x^2 +
x)'

 f'(x) = (e^x)' + (x^3)' - (x^2)' +
(x)'

 f'(x) =
e^x+3x^2-2x+1

We'll re-write the terms of the expression of
the first derivative, to complete the
squares:

f'(x)=e^x+2x^2+x^2-2x+1

We'll
combine the last 3 terms, because we've noticed that they are the result of squaring the
binomial
(x-1).

(a+b)^2=a^2+2ab+b^2

f'(x)=e^x+2x^2+(x-1)^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.