Consider the function Consider the function x-coordinate is (, ) Classification:

Question

Consider the function

Consider the function x-coordinate is (, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined) x-coordinate is (, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest x-coordinate is (, ) Classification: local minimum local maximum saddle point cannot be determined (local minimum, local maximum, saddle point, cannot be determined) The critical point with the next smallest f_{yy} = The critical point with the smallest f_{xy} = f_{xx} = f_y = f_x = f(x,y) = ysqrt x - y^2 - 5 x + 19 y. Find and classify all critical points of the function. If there are more blanks than critical points, leave the remaining entries blank.

Explanation / Answer

f(x, y) = yx1/2 - y2 - 5x + 19y
f_x = 1/2 yx-1/2 - 5
f_xy = 1/2x-1/2
f_xx = -1/4 yx-3/2
f_y = x1/2 - 2y + 19
f_yy = -2

for critical points, f_x = 0 and f_y = 0
f_x = 0 gives
1/2 yx-1/2 - 5 = 0
yx-1/2 = 10
y = 10x1/2.............(1)
f_y = 0 gives
x1/2 - 2y + 19 = 0
putting the value of (1) above
x1/2 - 2* 10x1/2 + 19 = 0
-19x1/2 + 19 = 0
x1/2 = 1
x = 1
at x = 1, y = 10

So the critical point is (1, 10)

D = f_xx * f_yy - (f_xy)2
D = (-1/4 yx-3/2)(-2) - (1/2x-1/2)2

D(1, 10) = 1/2 * 10 * 1 - 1/4
D (1/10) = 19/4 > 0
f_xx (1, 10) = -1/4 yx-3/2 = -1/4 * 10 * 1 = -5/2 < 0

Thus we have local maximum at (1, 10)
NO LOCAL MINIMUM
NO SADDLE POINT

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