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Consider the curve z = x4 - 2.x2. Write the curve parametrically in 3 dimensions. Using the linear transformation for rotation, rotate this curve around the z axis. Convert the parametric equation for the surface to an equation of the form z = f(x,y). Hint: make use of the identity cos2 theta + sin2 theta = 1. Find the critical points of the resulting surface. Determine the nature of the points using the second derivative test . If the test fails, explain why (this may take some research into the more technical definition of local extrema). Describe and carry out your own alternative method to determine the nature of the critical point/s for which the test fails.

Show transcribed image text Consider the curve z = x4 - 2.x2. Write the curve parametrically in 3 dimensions. Using the linear transformation for rotation, rotate this curve around the z axis. Convert the parametric equation for the surface to an equation of the form z = f(x,y). Hint: make use of the identity cos2 theta + sin2 theta = 1. Find the critical points of the resulting surface. Determine the nature of the points using the second derivative test . If the test fails, explain why (this may take some research into the more technical definition of local extrema). Describe and carry out your own alternative method to determine the nature of the critical point/s for which the test fails.