Find the critical points of the given function. Use the Second Derivative Test to determine if each critical point corresponds to a relative maximum, minimum, or saddle point.
\[f(x,y) = \dfrac{1}{2}x^2 + 2y^2 - 8y + 4x\]
Find the partial derivatives of f:
\[f_x(x, y) = x +4 ; \\ f_y(x,y) = 4y - 8\]
Set each equal to 0 and solve for x and y:
\[f_x(x, y) = 0 => x = -4 ; \\ f_y(x,y) = 0 => y = 2\]
Therefore, we have one critical point: (-4, 2). To determine if it corresponds to a relative maximum, minimum, or saddle point, we will look at the second partial derivative of f.
Let z = f(x, y) be differentiable on an open set containing P = (x_0 , y_0), and let
\[D = f_{xx}(x_0, y_0)f_{yy}(x_0, y_0) − f_{xy}^2(x_0, y_0)\]
If D > 0 and \[f_{xx}(x_0,y_0) > 0\], then P is a relative minimum of f
If D > 0 and \[f_{xx}(x_0,y_0) < 0\], then P is a relative maximum of f
If D<0, then P is a saddle point of f
If D = 0, the test is inconclusive
\[f_{xx} = 1 \\ f_{yy} = 4 \\ f_{xy} = 0\]
Therefore, \[D(x,y) = 4\] and (-4, 2) is a relative maximum of f.
What did you take away from this course?
This course was a good refresher of probability, calculus and linear algebra. Could have skipped linear regression, but was really good at getting us acquainted with many of the necessary Data Science skills.