1 Point) Let F(x)=|x₁| |x₄|. Use Interval Notation To Indicate The Values Of X Where F Is Differentiable.
Understanding the differentiability of functions involving absolute value expressions is fundamental in calculus, especially when analyzing piecewise functions and their smoothness properties. In this article, we will explore the function F(x) = |x₁| · |x₄|, where x = (x₁, x₂, x₃, x₄), and determine precisely where F is differentiable using interval notation. This comprehensive analysis will include detailed explanations of absolute value functions, their derivatives, and the points of non-differentiability, providing clarity for students and practitioners alike.
Understanding the Function F(x) = |x₁| · |x₄|
Breakdown of the Function Components
The function F(x) = |x₁| · |x₄| is a product of two absolute value functions:
- |x₁|, which depends solely on the first coordinate
- |x₄|, which depends solely on the fourth coordinate
The other components x₂ and x₃ do not influence the value of F(x), making the function effectively dependent on the first and fourth variables.
Properties of Absolute Value Functions
The absolute value function |x| is defined as:
|x| =
- x, if x ≥ 0
- -x, if x < 0
It is continuous everywhere but not differentiable at x = 0, because the derivative from the left and right do not coincide at zero.
Similarly, for the function F(x), the points where |x₁| or |x₄| are not differentiable are precisely where x₁ = 0 or x₄ = 0.
Differentiability of F(x) = |x₁| · |x₄|
Differentiability in Multivariable Context
In multivariable calculus, a function is differentiable at a point if all its partial derivatives exist and are continuous in a neighborhood around that point. For F(x), the partial derivatives with respect to each variable are:
- ∂F/∂x₁
- ∂F/∂x₂
- ∂F/∂x₃
- ∂F/∂x₄
Since F depends only on x₁ and x₄, the partial derivatives with respect to x₂ and x₃ are zero everywhere, and their differentiability is trivial.
The critical focus is on the partial derivatives with respect to x₁ and x₄, which involve the derivatives of absolute value functions.
Calculating Partial Derivatives
Let's compute the partial derivatives:
- Partial derivative with respect to x₁:
\[
\frac{\partial F}{\partial x1} = \frac{\partial}{\partial x1} (|x1| \cdot |x4|) = |x4| \cdot \frac{\partial}{\partial x1} |x_1|
\]
Since |x₁| is not differentiable at x₁ = 0, its derivative is:
\[
\frac{d}{dx1} |x1| =
\begin{cases}
1, & x_1 > 0 \\
-1, & x_1 < 0 \\
\text{undefined}, & x_1 = 0
\end{cases}
\]
Therefore,
\[
\frac{\partial F}{\partial x_1} =
\begin{cases}
|x4|, & x1 \neq 0 \\
\text{undefined}, & x_1 = 0
\end{cases}
\]
- Partial derivative with respect to x₄:
Similarly,
\[
\frac{\partial F}{\partial x4} = |x1| \cdot \frac{\partial}{\partial x4} |x4| = |x_1| \times
\begin{cases}
1, & x_4 > 0 \\
-1, & x_4 < 0 \\
\text{undefined}, & x_4 = 0
\end{cases}
\]
Thus,
\[
\frac{\partial F}{\partial x_4} =
\begin{cases}
|x1|, & x4 \neq 0 \\
\text{undefined}, & x_4 = 0
\end{cases}
\]
Points of Non-Differentiability
The partial derivatives exist and are finite everywhere except where the derivatives of the absolute value functions are undefined, specifically at x₁ = 0 and x₄ = 0.
Therefore, the function F(x) is not differentiable at points where:
- x₁ = 0
- x₄ = 0
Since the differentiability depends on the behavior of the absolute value functions, the overall set of points where F is not differentiable is the union of the hyperplanes:
- { (x₁, x₂, x₃, x₄) | x₁ = 0 }
- { (x₁, x₂, x₃, x₄) | x₄ = 0 }
Conversely, F is differentiable at all points where both x₁ ≠ 0 and x₄ ≠ 0.
Using Interval Notation to Express Differentiability
In multivariable calculus, the set of points where a function is differentiable can be described using Cartesian product notation. For the variables x₁ and x₄:
- The set where F is differentiable with respect to these variables is:
\[
\{ (x1, x4) \mid x1 \neq 0, x4 \neq 0 \}
\]
Expressed in interval notation for each variable:
- For x₁:
\[
(-\infty, 0) \cup (0, \infty)
\]
- For x₄:
\[
(-\infty, 0) \cup (0, \infty)
\]
Therefore, the set of points where F is differentiable can be written as:
\[
\left( (-\infty, 0) \cup (0, \infty) \right) \times \left( (-\infty, 0) \cup (0, \infty) \right)
\]
In terms of the entire 4-dimensional space:
\[
\{ (x1, x2, x3, x4) \mid x1 \in (-\infty, 0) \cup (0, \infty), \quad x4 \in (-\infty, 0) \cup (0, \infty) \}
\]
Since x₂ and x₃ do not affect differentiability, they can take any real values.
Summary of Differentiability Regions
Based on the above analysis, the function F(x) = |x₁| · |x₄| is differentiable exactly where:
- x₁ ≠ 0
- x₄ ≠ 0
Expressed in interval notation:
\[
\boxed{
\left( (-\infty, 0) \cup (0, \infty) \right) \times \mathbb{R} \times \mathbb{R} \times \left( (-\infty, 0) \cup (0, \infty) \right)
}
\]
or, focusing solely on x₁ and x₄:
\[
\boxed{
\left( (-\infty, 0) \cup (0, \infty) \right) \times \left( (-\infty, 0) \cup (0, \infty) \right)
}
\]
which indicates that F is differentiable for all points where x₁ and x₄ are non-zero, regardless of the values of x₂ and x₃.
Conclusion
In conclusion, the differentiability of the function F(x) = |x₁| · |x₄| hinges on the behavior of the absolute value functions involved. The points of non-differentiability are precisely where either x₁ or x₄ equals zero. Using interval notation, the set of points where F is differentiable can be precisely characterized as:
\[
\boxed{
\left( (-\infty, 0) \cup (0, \infty) \right) \times \mathbb{R} \times \mathbb{R} \times \left( (-\infty, 0) \cup (0, \infty) \right)
}
\]
This analysis underscores the importance of understanding the subtleties of absolute value functions within multivariable calculus and provides a clear framework for identifying differentiability regions in higher dimensions.
