The function f(x) = x + cos x is

The correct option is always increasing.

To determine whether the function f(x)=x+cosx is increasing or decreasing, we need to analyze its first derivative with respect to x. Recall that a function is strictly increasing on an interval if its derivative is strictly positive for all values of x in that interval (or non-negative and not constantly zero on any sub-interval).

First, let's find the derivative of f(x):
f'(x)=ddx(x+cosx)
Using the sum rule and standard derivatives, we get:
f'(x)=1-sinx

Next, we analyze the behavior of the expression 1-sinx. We know that the sine function, sinx, is bounded for all real values of x by the inequality:
-1≤sinx≤1

Multiplying the entire inequality by -1 reverses the inequality signs:
1≥-sinx≥-1
which can be rewritten as:
-1≤-sinx≤1

Now, add 1 to all parts of the inequality:
1-1≤1-sinx≤1+1
Simplifying this yields:
0≤1-sinx≤2

Since f'(x)=1-sinx, we have:
f'(x)≥0
for all real values of x.

The derivative f'(x) becomes zero only at isolated points where sinx=1 (i.e., at x=π2+2nπ where n is an integer). Because these points are isolated and the derivative is strictly positive elsewhere, the function does not remain constant over any interval. Therefore, the function f(x) is strictly increasing for all real values of x.