f(x,y) = (x^2 - 1)^2 + (y - 4)^2 + 1 Find all critical points of f and classify each as a local...


{eq}f(x,y) = (x^2 - 1)^2 + (y - 4)^2 + 1 {/eq}

Find all critical points of f and classify each as a local max, min, or saddle points.

Critical Points:

We have a function which has a square of a quadratic function and a square of a linear function. We will first find the critical points and then we will classify them using the second derivative test.

{eq}D=f_{xx}f_{yy}-(f_{xy})^2\\ D>0,f_{xx}>0~\text{the point is local minimum}\\ D>0,f_{xx}<0~\text{the point is local maximum}\\ D<0~\text{the point is a saddle point} {/eq}

Answer and Explanation:

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{eq}f(x,y) = (x^2 - 1)^2 + (y - 4)^2 + 1 {/eq}

We will find the critical points with the help of the partial derivatives:

{eq}f_{x}=4x(x^2-1)\\ f_...

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Learn more about this topic:

Partial Derivative: Definition, Rules & Examples


Chapter 18 / Lesson 12

When a function depends on more than one variable, we can use the partial derivative to determine how that function changes with respect to one variable at a time. In this lesson, we use examples to define partial derivatives and to explain the rules for evaluating them.

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