Which of the following function is a monotonically increasing function?
f(x)=x3+2x
We saw that for a monotonically increasing function f’(x) ≥ 0. We will go through the options to find the answer.
a. f(x) = x2+2x
The domain of this function is R.
f’(x) = 2x+2=2(x+1)
2x+2 will be non negative or greater than equal to zero for x≥−1. For x<−1, f’(x) will be negative.
So the function will be increasing only in the interval of [−1,∞) and will be decreasing in the rest of the domain. Thus the function is not monotonically increasing function
b. f(x) = x3+2x
The domain for this function is also R.
f’(x) = 3x2+2
3x2+2≥0 for all real numbers.
So the function will be monotonically increasing in its domain.
Derivative of sinx would give cosx and derivative of cosx would give -sinx. Since -sinx and cosx takes both positive and negative values, we can’t say both the options C and D behaves the same way throughout their domain. So, they can’t be monotonically increasing as well.
{Note - If specifically not mentioned we’ll consider all those points in the domain where the functions exists. }