Q.185. Critical points of a two variables function

Question 185.
Find the critical points of the function f(x,y)=6x^7+7y^2+8xy+9.

Answer 185.
To find critical points of a function of two variables, find partial derivative of the function with respect to each variable and set to zero. Two equations, thus, obtained should be solved to obtain the critical points.

Partial derivative of f(x, y) = 6x^7 + 7y^2 + 8xy + 9 with respect to x,
∂[f(x, y)]/∂x = 0
=> 42x^6 + 8y = 0 ... (1)

Partial derivative of f(x, y) = 6x^7 + 7y^2 + 8xy + 9 with respect to y,
∂[f(x, y)]/∂y = 0
=> 14y + 8x = 0 ... (2)

Solving equations (1) and (2) for x by eliminating y,
42x^6 + 8(-8x/14) = 0
=> 147x^6 - 16x = 0
=> x [147x^5 - 16] = 0
=> x = 0 or (16/147)^(1/5)
and y = 0 or (-4/7) * (16/147)^(1/5)

=> Critical points are (0, 0) or (0.6417, - 0.3667).

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