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Questions › GATE › GATE 2026 › Q60
2 marksMathematics › Multivariable calculus › Implicit functions, level curves and second-order conditionsHardabout 3 min

GATE 2026 Question 60: Multivariable calculus

Consider the function G(x,y,z)=0G(x,y,z)=0G(x,y,z)=0. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function G(x,y,z)G(x,y,z)G(x,y,z) exist everywhere. Then, the value of (∂z∂x×∂x∂y×∂y∂z)\left(\dfrac{\partial z}{\partial x}\times\dfrac{\partial x}{\partial y}\times\dfrac{\partial y}{\partial z}\right)(∂x∂z​×∂y∂x​×∂z∂y​) is ______ (in integer)

Answer and worked solution

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Plain-text version

Consider the function G(x,y,z)=0. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function G(x,y,z) exist everywhere. Then, the value of ((∂ z)/(∂ x)×(∂ x)/(∂ y)×(∂ y)/(∂ z)) is (in integer)

Also written as: Consider the function G(x,y,z)=0. This function allows us to implicitly define each of three variables as a function of the other two variables. Assume that all partial derivatives of the function G(x,y,z) exist everywhere. Then, the value of ((d z)/(d x)*(d x)/(d y)*(d y)/(d z)) is (in integer)

Multivariable calculus has come up 32 times in 7 exams since 2007.

What this question tests

This question is on Implicit functions, level curves and second-order conditions, part of Multivariable calculus in Mathematics.

Slopes of level curves and implicit functions, and conditions for optima in two variables.

TestStop has 8 past-paper questions on this subtopic.

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