If fx Greater Than Zero is f Strictly Increasing?

Understanding the relationship between the derivative of a function and its monotonicity is crucial in calculus and analysis. Specifically, if fx’ > 0 for a function for all real values of x, then fx is called monotonically increasing. However, if fx’ 0 for all real x, then the function is said to be non-decreasing.

Monotonicity and the Derivative

If the derivative of any function, say f(x), is strictly positive at any point, it follows from the definition of the derivative that in some neighborhood of that point, the function is monotonically increasing. This means that fx’ > 0 implies that the slope of the function is positive, thus the function is increasing in that neighborhood.

More formally, if fx’ > 0 in a certain interval, it means that the function fx is increasing in that interval. Conversely, if fx is increasing in any interval, it is necessary that fx’ > 0 in that interval.

Exceptions and Counterexamples

It is important to note that there are exceptions to the above rules. Specifically, if the derivative of a function cannot be strictly positive for all real values of x, the function may not be monotonically increasing. For instance, if fx0 for R only if fx is linear, such as fx ax b. However, if fx has higher degree polynomials, its domain would be restricted, which can lead to the function not being strictly increasing.

Another key point is that a function can be strictly increasing without its derivative being strictly positive everywhere. For example, the arctangent function, arctan(x), has a derivative that is positive but not strictly greater than zero everywhere, yet it is still a strictly increasing function.

Counterexample: arctan(x)

To illustrate this, consider the function fx arctan(x). The derivative of arctan(x) is given by:

fx’ 1 / (1 x^2)

While 1 / (1 x^2) is always positive, it is not strictly greater than zero. Yet, arctan(x) is still a strictly increasing function. This shows that a function can be increasing without its derivative being strictly positive everywhere.

Additional Considerations

When dealing with functions that are strictly increasing or monotonically increasing, it is essential to consider the absence of non-removable discontinuities. If a function has no non-removable discontinuities, then the relationship between the derivative and monotonicity holds more consistently. However, as pointed out by Larry, this is not a universal rule and there are valid counterexamples.

In conclusion, while a strictly positive derivative elsewhere implies that a function is monotonically increasing, there are exceptions and nuances to consider, particularly when evaluating functions that are not linear or free from non-removable discontinuities.