Show that function f(x)=2x^3+6x-5 has not any extremes!

For a function to allow a local extreme , it's derivative
has to have real roots.

We'll determine the derivative of
the function:

f'(x) = 6x^2 +
6

6x^2 + 6 > 0, for any real
x

It is obvious that the equation is not cancelling for any
real value of x, so the 1st derivative has no real
roots.

Since the derivative of the function
is strictly positive, the function does not allow local
extremes.