To determine the behavior of the function
f(x)=cosx−1+ for
x∈ℝ, we need to analyze its first derivative. The first derivative of a function helps us understand whether the function is increasing or decreasing.
Let's compute the first derivative,
f′(x), of the given function:
f(x)=cosx−1+The derivative of
cosx is
−sinx. The derivative of a constant -1 is zero. The derivative of
is
=x. Therefore:
f′(x)=−sinx+xTo analyze the behavior of
f(x), we need to examine the sign of
f′(x) (i.e., whether it is positive or negative).
If
f′(x)>0, the function is increasing. If
f′(x)<0, the function is decreasing.
Firstly, consider the function for
x∈ℝ.
There is no interval where the sign of
f′(x) is always positive or always negative. This is due to the oscillatory nature of the
−sinx component and the linear component
x. The increasing linear component
x and the oscillating
−sinx means the derivative
f′(x) does not maintain a consistent sign over the entire real line.
Therefore,
f(x) is neither a purely increasing function nor a purely decreasing function over the entire domain.
Thus, the correct option is:
Option C: neither increasing nor decreasing