Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function.
A)(4, 3); saddle point: f(4, 3) = 66
B)(- 3, 4); relative maximum value: f(- 3, 4) = - 9
C)(3, - 4); relative minimum value: f(3, - 4) = - 66

Question

Find the critical point(s) of the function.Then use the second derivative test to classify the nature of each point, if possible.Finally, determine the relative extrema of the function. 
A)(4, 3); saddle point: f(4, 3) = 66 
B)(- 3, 4); relative maximum value: f(- 3, 4) = - 9 
C)(3, - 4); relative minimum value: f(3, - 4) = - 66

Quizplus · Accepted Answer

We can use the second derivative test to classify the critical points: For point A: fxx(4,3) = 8 > 0 (Minimum) fxy(4,3) = -6 fyy(4,3) = 18 > 0 The discriminant is: D = fxx(4,3)fyy(4,3) - [fxy(4,3)]^2 = 8*18 - (-6)^2 = 144 Since D > 0 and fxx(4,3) > 0, we have a relative minimum at point A. For point B: fxx(-3,4) = 10 > 0 (Minimum) fxy(-3,4) = -4 fyy(-3,4) = -8 < 0 The discriminant is: D = fxx(-3,4)fyy(-3,4) - [fxy(-3,4)]^2 = 10*(-8) - (-4)^2 < 0 Since D < 0, we have a saddle point at point B. For point C: fxx(3,-4) = -8 < 0 (Maximum) fxy(3,-4) = -6 fyy(3,-4) = 10 > 0 The discriminant is: D = fxx(3,-4)fyy(3,-4) - [fxy(3,-4)]^2 = (-8)*10 - (-6)^2 < 0 Since D < 0 and fxx(3,-4) < 0, we have a relative maximum at point C. Therefore, the only correct choice is C: (3,-4); relative minimum value: f(3,-4) = -66.

Find the Critical Point(s) of the Function