Question

Prove that, if \(f^{\prime}(x)\) exists and is continuous on an interval \(I\) and if \(f^{\prime}(x) \neq 0\) at all interior points of \(I\), then either \(\bar{f}\) is increasing throughout \(I\) or decreasing throughout \(I .\) Hint: Use the Intermediate Value Theorem to show that there cannot be two points \(x_{1}\) and \(x_{2}\) of \(I\) where \(f^{\prime}\) has opposite signs.

Short answer

If \( f'(x) \neq 0 \) and is continuous on \( I \), then \( f(x) \) is consistently increasing or decreasing on \( I \).

Step-by-step solution

Assumptions and Definition

Assume that the function \( f(x) \) is continuously differentiable on the interval \( I \), meaning \( f'(x) \) exists and is continuous on \( I \). Additionally, \( f'(x) eq 0 \) at all interior points of \( I \).

Applying the Intermediate Value Theorem (IVT)

The Intermediate Value Theorem states that if a function \( h(x) \) is continuous on an interval \([a, b]\) and \( k \) is a number between \( h(a) \) and \( h(b) \), then there exists at least one \( c \) in \( [a, b] \) such that \( h(c) = k \). Here, \( h(x) = f'(x) \) is continuous, and \( f'(x) \) never equals zero; hence, it must maintain the same sign throughout the interval.

Argument by Contradiction

Assume for contradiction that there exist two points \( x_1 \) and \( x_2 \) in \( I \) such that \( f'(x_1) > 0 \) and \( f'(x_2) < 0 \). By IVT, since \( f'(x) \) is continuous, there must be some point \( c \) between \( x_1 \) and \( x_2 \) where \( f'(c) = 0 \), contradicting \( f'(x) eq 0 \).

Conclusion

Since it is impossible for \( f'(x) \) to change signs while still being continuous and nonzero, \( f'(x) \) must retain a consistent sign across the entire interval \( I \). Consequently, \( f(x) \) is either entirely increasing (\( f'(x) > 0 \)) or entirely decreasing (\( f'(x) < 0 \)) on \( I \).

Key concepts

Continuously Differentiable

When we talk about a function being continuously differentiable on an interval, we're referring to two related concepts:

• The derivative of the function exists at every point within the interval.
• This derivative is continuous throughout that interval.

In simple terms, for a function to be continuously differentiable, it should not only have a derivative at each point but that derivative should not "jump" or have breaks.
Imagine a smooth curve that you can glide along without interruption. That's what a continuously differentiable function looks like.
This property is important because it guarantees that there won't be any sudden changes in the behavior of the derivative, which is essential for applying the Intermediate Value Theorem.
For example, if you're computing the slope at different points on a road and it's continuously differentiable, it implies a smooth journey with no unexpected sharp turns or stops.

Derivative Sign

The sign of the derivative, whether positive or negative, tells us a lot about how the function behaves in an interval.

• A positive derivative indicates that the function is increasing, as the function's values rise as you move along x-axis.
• A negative derivative indicates a decreasing function, where the function's values fall as you progress.

Now imagine if you are hiking up a hill. When you are going up, your elevation is increasing, which would be represented by a positive derivative.
Conversely, when you come back down, your elevation decreases, indicating a negative derivative.
Thus, if a derivative does not equal zero and maintains the same sign, the function won't flip directions or "slope."
Learn more about derivatives and sign. This characteristic is crucial in understanding how a function behaves over a given interval, as it can tell us if the function is steadily rising or falling without any wavering.

Increasing and Decreasing Functions

The determination of whether a function is increasing or decreasing over an interval can be made by inspecting the sign of its derivative.
A continuously differentiable function that maintains a constant positive derivative through an interval is said to be increasing in that interval. This simply means that as the value of x increases, the value of the function also increases.
Conversely, if the function's derivative is consistently negative, then the function is decreasing on that interval.
The link between the derivative sign and function characteristic is validated by the theorem proven in the exercise. Here, if a function's derivative does not cross zero, it ensures the derivative does not switch from positive to negative (or vice versa)—which means there can't be a "peak" or "valley" within the interval.
So, with the condition that the derivative is never zero, an increasing or decreasing behavior without change throughout the interval is guaranteed.
Understanding this concept helps to picture the function as either a consistently ascending or descending path on a graph.