There exists a function f(x,y) where contour lines associated with two different constants k can intersect.
True
False

The statement is **false**.

Contour lines of a function \( f(x, y) \) correspond to curves where the function takes a constant value, i.e., \( f(x, y) = k_1 \) for one contour and \( f(x, y) = k_2 \) for another. If two contour lines intersect, it would mean that at the point of intersection, the function takes two different values simultaneously, which is impossible for a well-defined function.

Thus, contour lines associated with different constants cannot intersect.

Would you like more details or examples?

Here are 5 related questions to explore this concept:
1. What are contour lines in multivariable calculus?
2. How do gradient vectors relate to contour lines?
3. Can a contour line for a single constant have multiple branches?
4. How do you determine the shape of contour lines for a specific function?
5. What is the significance of the level set in multivariable functions?

**Tip:** The gradient of a function is always perpendicular to its contour lines at any point.

There exists a function f(x,y) where contour lines associated with two different constants k can intersect. True or False?

Explore whether contour lines of a function f(x, y) associated with two different constants can intersect. This problem delves into concepts of multivariable calculus, functions of two variables, and level curves. The solution provides a clear explanation and examples, suitable for undergraduate calculus students.

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