Question

Prove or give a counterexample: If \(f\) is differentiable and \(f_{x}=0\) in a region \(D\), then \(f\left(x_{1}, y\right)=f\left(x_{2}, y\right)\) whenever \(\left(x_{1}, y\right)\) and \(\left(x_{2}, y\right)\) are in \(D ;\) that is \(f(x, y)\) depends only on \(y\).

Short answer

#Answer# In the given scenario, since the function \(f(x, y)\) is differentiable in region D and its partial derivative with respect to x, \(f_x\), is equal to zero in D, it implies that the function only depends on the y-coordinate in region D. This means that for any two points \((x_1, y)\) and \((x_2, y)\) in D, we have \(f(x_1, y) = f(x_2, y)\).

Step-by-step solution

1. Differentiability and partial derivatives

A function \(f(x, y)\) is differentiable in region D if it has continuous partial derivatives in that region. In this case, we have two partial derivatives: \(f_x\) and \(f_y\). Since the function is differentiable, both partial derivatives exist and are continuous in D.

2. Use the given condition

We are given that \(f_x = 0\) in region D. This means that the rate of change of the function with respect to the x-axis (x-coordinate) is zero at every point in region D.

3. Apply the Mean Value Theorem for partial derivatives

Since \(f_x = 0\) and the function is differentiable, we can apply the Mean Value Theorem. For any two points \((x_1, y)\) and \((x_2, y)\) in region D, there exists a point \((c, y)\), such that \(x_1 < c < x_2\), and: $$f_x(c,y)(x_2-x_1) = f(x_2, y) - f(x_1, y)$$

4. Show the difference between function values is zero

According to the given condition, \(f_x(c, y) = 0\) for any point \((c, y)\) in region D. Therefore, we can rewrite the above equation: $$0 \cdot (x_2 - x_1) = f(x_2, y) - f(x_1, y)$$ This simplifies to: $$f(x_2, y) - f(x_1, y) = 0$$ Which gives us: $$f(x_2, y) = f(x_1, y)$$ This shows that the function \(f(x, y)\) depends only on the y-coordinate in region D, as required.

Key concepts

Mean Value Theorem

The Mean Value Theorem (MVT) for multivariable functions, like the one described in our exercise, is a powerful tool in understanding how a function behaves between two points. In essence, MVT states that for a differentiable function there exists at least one point where the instantaneous rate of change (the derivative) is the same as the average rate of change over the interval.

Specifically, for a function of two variables, such as our function f(x, y), if it has continuous partial derivatives and we consider two points in a region D that have the same y-value but different x-values, there will be at least one point where the partial derivative with respect to x is equal to the average change of f with respect to x. This principle lays the foundation for the understanding of our exercise, which involves proving that the function's value does not change along the x-axis, implying it depends only on y.

Differentiable Functions

Differentiation is a fundamental concept in calculus that deals with how we calculate the rate at which a quantity changes. Functions that can be differentiated are called differentiable functions. In the context of functions of two variables f(x, y), being differentiable means that the function has a well-defined tangent plane at each point within the domain, and tiny changes in x and y result in linear changes to the function's output when approximated closely.

It's crucial for students to understand that differentiability implies continuity; however, the converse is not necessarily true. For partial derivatives to exist and be continuous as mentioned in Step 1 of the solution, the function must be smooth enough to allow for a clear linear approximation in the vicinity of each point within region D. Such properties ensure that the conclusions we draw from the Mean Value Theorem hold true.

Rate of Change

In calculus, the term 'rate of change' reflects how a function's output varies as its input changes. In single-variable functions, this is simply the derivative. For functions of several variables, such as f(x, y), rate of change can be described by partial derivatives, which measure how f changes as either x or y is varied individually.

For instance, the partial derivative f_x represents the rate of change of f with respect to x, holding y constant. In the exercise, having f_x = 0 implies the function's value is unaltered, no matter how x fluctuates; mathematically, this indicates a horizontal slope when moving along the x-axis. This concept is vital to identifying the function's behavior in region D and confirming that f(x, y) does not depend on x there.

Continuous Partial Derivatives

Continuity in partial derivatives is an extension of the concept of continuity from single-variable calculus to functions of multiple variables. In our exercise, continuous partial derivatives mean that the function's rate of change with respect to each variable smoothly transitions from point to point without any abrupt jumps or breaks within the region D.

When a function f(x, y) has continuous partial derivatives, it signifies that small changes in x or y lead to small changes in the derivatives. This smoothness is an essential condition for applying the Mean Value Theorem to functions of several variables. The continuous nature of f_x and f_y guarantees that the function behaves predictably and that tools like the MVT can be applied to analyze the function's profile in an entire region, further refining our understanding of a function's dependency on its variables. For students, getting a grip on this concept assures them processing subsequent calculus problems with greater confidence.