If ∂f∂x=0∂𝑓∂𝑥=0, the function f(x,y)𝑓(𝑥,𝑦) has no dependence on the variable x𝑥.Select one:TrueFalse

Step 1: Break Down the Problem

We need to determine the implications of the statement $\frac{\partial f}{\partial x} = 0$ for the function $f(x,y)$. This involves understanding the meaning of the partial derivative with respect to $x$.

Step 2: Relevant Concepts

The statement $\frac{\partial f}{\partial x} = 0$ indicates that the rate of change of $f$ with respect to $x$ is zero. This suggests that varying $x$ does not change $f(x,y)$, indicating that $f$ is independent of $x$.

Step 3: Analysis and Detail

1. A partial derivative $\frac{\partial f}{\partial x}$ computes how $f$ changes as $x$ changes, while keeping $y$ constant.
2. If $\frac{\partial f}{\partial x} = 0$, it means that changes in $x$ do not affect the value of the function $f$.
3. Therefore, $f(x,y)$ must be a function solely of $y$ (i.e., $f$ does not vary with $x$).

Step 4: Verify and Summarize

Given that the partial derivative with respect to $x$ is zero, the function $f(x,y)$ cannot depend on $x$. This implies that the statement is indeed true.

Final Answer

True