## Question

Investigate the nature of critical points for the given functions:

a)

b)

## Solution

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### a)

__Step-1: Find critical points:__

*Thus, (2, 0) is the critical point for f(x, y).*

__Step-2: Find Eigenvalues at critical point:__

Hessian matrix of f,

__Step-3: Find nature of the critical point:__

*Since eigenvalues of Hf(2,0) are positive, (2, 0) is a local minimum.*

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### b)

__Step-1: Find critical points:__

(1)

(2)

(3)

(4)

Similarly,

(5)

(6)

(7)

(7) in (1)

From (7), we get

*Thus, is the critical point for f(x, y).*

__Step-2: Find Eigenvalues at critical point:__

Hessian matrix of f, Hf(x,y)=∣∣∣∂2f/∂x2∂2f/∂x∂y∂2f/∂x∂y∂2f/∂y2∣∣∣ = ∣∣∣x+3/x311y+3/y3∣∣∣

__Step-3: Find nature of the critical point:__

*Since eigenvalues of are positive, is a local minimum.*