Given:- Function f(x) = e2x
Theorem:- Let f be a differentiable real function defined on an open interval (a, b).
(i) If f’(x) > 0 for all x ∈ (a, b), then f(x) is increasing on (a, b)
(ii) If f’(x) < 0 for all x ∈ (a, b), then f(x) is decreasing on (a, b)
Algorithm:-
(i) Obtain the function and put it equal to f(x)
(ii) Find f’(x)
(iii) Put f’(x) > 0 and solve this inequation.
For the value of x obtained in (ii) f(x) is increasing and for remaining points in its domain it is decreasing.
Here we have,
For f(x) to be increasing, we must have
since, the value of e lies between 2 and 3
so, whatever be the power of e (i.e x in domain R) will be greater than zero.
Thus f(x) is increasing on interval R
