Question

Let Z = f(x,y) = 3x3 -5y2 -225x + 70y + 23. (i) Find the stationary points of z. (ii) Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point. 

Answer 1

Given f(x, y) =  3x³ - 5y² - 225x + 70y + 23

(i) for stationary point: f_x = 0 & f_y = 0

$\implies$ f_x(x, y) = 9x² - 225 = 0

9x² = 225

x² = $\frac{225}{9}$

x = $\sqrt{\frac{225}{9}}$

x = $\pm$ $\frac{15}{3}$

$\implies$ f_y(x, y) = -10y + 70 = 0

-10y = -70

y = $\frac{70}{10}$

y = 7

therefore the possible stationary points are ( $\frac{15}{3}$, 7) and ( -$\frac{15}{3}$, 7)

(ii) now to Determine if at these points the function is at a relative maximum, relative minimum, infixion point, or saddle point. 

f_xy = 0 , f_xx = 18x and f_yy = -10

therefore,

D = f_xx$\cdot$ f_yy - (f_xy)² = (18x)(-10) - (0)² = -180x

D( $\frac{15}{3}$, 7) = - 180 $\cdot \frac{15}{3}$ = -900 < 0

D( - $\frac{15}{3}$, 7) = - 180 $\cdot \frac{-15}{3}$ = 900 > 0

f_xx( - $\frac{15}{3}$, 7) = 18 $\cdot \frac{-15}{3}$ = -90 < 0

therefore by second order extremum test,

$\implies$ ( -$\frac{15}{3}$, 7) is relative maximum and ( $\frac{15}{3}$, 7) is saddle point