\(\displaystyle z= 3x^2 + 7x + 2y^2 - 5y -xy + 13\)

Find the stationary point and determine the nature of the point(g).

My question is did any step im do wrongly?

STATIONARY POINT

\(\displaystyle \frac{\partial z}{\partial x} = 6x + 7 - y\)

\(\displaystyle \frac{\partial z}{\partial y} = 4y - 5 - x\)

\(\displaystyle \frac{\partial z}{\partial x} = 0 ,\frac{\partial z}{\partial y} = 0\)

\(\displaystyle 6x + 7 - y = 0\)

\(\displaystyle 4y - 5 - x = 0\)

Then x and y value is:

\(\displaystyle x = \frac{-3}{19}\)

\(\displaystyle y = \frac{23}{19}\)

Substitute x and y into original equation(z), then get \(\displaystyle z=\frac{3263}{361}\)

stationary point is at (\(\displaystyle \frac{-3}{19}\),\(\displaystyle \frac{23}{19}\),\(\displaystyle \frac{3263}{361}\))

Now i want find the nature of the point(g):

\(\displaystyle g = (\frac{\partial^2 z}{\partial x^2})(\frac{\partial^2 z}{\partial y^2}) -(\frac{\partial^2 z}{\partial x\partial y} )^2 \)

\(\displaystyle = 6(4)-(-1)^2\)

\(\displaystyle =23>0\)

Stationary point is a minimum point

Find the stationary point and determine the nature of the point(g).

My question is did any step im do wrongly?

STATIONARY POINT

\(\displaystyle \frac{\partial z}{\partial x} = 6x + 7 - y\)

\(\displaystyle \frac{\partial z}{\partial y} = 4y - 5 - x\)

\(\displaystyle \frac{\partial z}{\partial x} = 0 ,\frac{\partial z}{\partial y} = 0\)

\(\displaystyle 6x + 7 - y = 0\)

\(\displaystyle 4y - 5 - x = 0\)

Then x and y value is:

\(\displaystyle x = \frac{-3}{19}\)

\(\displaystyle y = \frac{23}{19}\)

Substitute x and y into original equation(z), then get \(\displaystyle z=\frac{3263}{361}\)

stationary point is at (\(\displaystyle \frac{-3}{19}\),\(\displaystyle \frac{23}{19}\),\(\displaystyle \frac{3263}{361}\))

Now i want find the nature of the point(g):

\(\displaystyle g = (\frac{\partial^2 z}{\partial x^2})(\frac{\partial^2 z}{\partial y^2}) -(\frac{\partial^2 z}{\partial x\partial y} )^2 \)

\(\displaystyle = 6(4)-(-1)^2\)

\(\displaystyle =23>0\)

Stationary point is a minimum point

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