Demonstrate that the function f(x)=e^2x+x^3-2x^2+4x is monotonically increasing.

Since we have to use the first derivative to verify if the
function is monotonic, first we need to check if the function is continuous. Since the
given function is an algebraic sum of elementary continuous functions, then f(x) is
continuous.

In order to decide the monotony of  f(x), we'll
have to demonstrate that the first derivative does not change its
sign.

We'll compute
f'(x)=2e^2x+3x^2-4x+4

We'll re-write
f'(x):

f'(x)=2e^2x+2x^2+x^2-4x+4

We
notice that the sum of the last 3 terms represents a perfect
square:

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

x^2-4x+4
= (x-2)^2

We'll re-write
f'(x):

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

Since
(x-2)^2>0 and 2e^2x+2x^2 > 0, then
2e^2x+2x^2+(x-2)^2>0.

We notice that
the expression of f'(x) keeps it's positive sign, for any value of x, therefore f(x) is
a monotonically increasing function.