For-x-y-R-find-the-minimum-and-maximum-of-2x-2-3x-4y-if-x-2-2y-2-xy-5x-7y-30-0-

Question Number 79417 by mr W last updated on 25/Jan/20

F o r x, y \in R f i n d t h e m i n i m u m a n d

m a x i m u m o f 2 x^{2} - 3 x + 4 y

i f x^{2} + 2 y^{2} - x y - 5 x - 7 y - 30 = 0 .

Commented by john santu last updated on 25/Jan/20

w h a t t h e a n s w e r ?

Commented by mr W last updated on 25/Jan/20

{i t}^{'} s n o t w o r k e d o u t y e t . i h a v e o n l y

a n i d e a, m a y b e n o t a g o o d o n e .

b a s i c a l l y i w a n t e d t o k n o w w h i c h

m e t h o d s w e c a n a p p l y t o s o l v e . {w h a t}^{'} s

y o u r s u g g e s t i o n s i r ?

Commented by john santu last updated on 25/Jan/20

y e s s i r, I a l s o {h a v e n}^{'} t s o l v e d

i t y e t . w i t h t h e L a g r a n g e

m u l t i p l i e r m e t h o d, i {c a n}^{'} t y e t

Answered by mr W last updated on 26/Jan/20

=== G e n e r a l w a y t o s o l v e ===

i t i s t o f i n d t h e m i n i m u m a n d

m a x i m u m o f t h e f u n c t i o n

F (x, y) = 2 x^{2} - 3 x + 4 y

u n d e r t h e c o n d i t i o n t h a t

G (x, y) = x^{2} + 2 y^{2} - x y - 5 x - 7 y - 30 = 0

U s i n g L a g r a n g e m u l t i p l i e r m e t h o d

L (x, y) = 2 x^{2} - 3 x + 4 y + λ (x^{2} + 2 y^{2} - x y - 5 x - 7 y - 30)

\frac{\partial L}{\partial x} = 4 x - 3 + λ (2 x - y - 5) = 0 \dots (i)

\frac{\partial L}{\partial y} = 4 + λ (4 y - x - 7) = 0 \dots (i i)

\frac{\partial L}{\partial λ} = x^{2} + 2 y^{2} - x y - 5 x - 7 y - 30 = 0 \dots (i i i)

f r o m (i) a n d (i i) :

\frac{3 - 4 x}{2 x - y - 5} = \frac{4}{x - 4 y + 7}

3 x - 12 y + 21 - 4 x^{2} + 16 x y - 28 x = 8 x - 4 y - 20

4 x^{2} - 16 x y + 33 x + 8 y - 41 = 0 \dots (i v)

x^{2} + 2 y^{2} - x y - 5 x - 7 y - 30 = 0 \dots (i i i)

w e g e t f r o m (i i i) a n d (i v) f o l l o w i n g

s o l u t i o n p o i n t s (n u m e r i c a l l y) :

(x, y) = (- 3.5386, 1.6666) \Rightarrow F = 42.3256

(x, y) = (0.1194, 6.0763) \Rightarrow F = 23.9755

(x, y) = (11.3374, 4.8863) \Rightarrow F = 242.6063

(x, y) = (0.7960, - 2.5756) \Rightarrow F = - 11.4232

\Rightarrow f_{m i n} = - 11.4232

\Rightarrow f_{m a x} = 242.6063

Commented by john santu last updated on 25/Jan/20

i h a v e d o n e i t l i k e t h i s, b u t i t

w a s c o n t r a i n e d b y t h e n u m e r i c a l

m e t h o d

Answered by mr W last updated on 27/Jan/20

=== A n o t h e r w a y t o s o l v e ===

x^{2} - (y + 5) x + 2 y^{2} - 7 y - 30 = 0

Δ = {(y + 5)}^{2} - 4 (2 y^{2} - 7 y - 30) ⩾ 0

\Rightarrow 7 y^{2} - 38 y - 145 ⩽ 0

y_{1, 2} = \frac{19 \pm 4 \sqrt{86}}{7}

\Rightarrow \frac{19 - 4 \sqrt{86}}{7} ⩽ y ⩽ \frac{19 + 4 \sqrt{86}}{7}

\Rightarrow y_{m} = \frac{19}{7}

2 y^{2} - (x + 7) y + x^{2} - 5 x - 30 = 0

Δ = {(x + 7)}^{2} - 8 (x^{2} - 5 x - 30) ⩾ 0

\Rightarrow 7 x^{2} - 54 x - 289 ⩾ 0

x_{1, 2} = \frac{27 \pm 8 \sqrt{43}}{7}

\Rightarrow \frac{27 - 8 \sqrt{43}}{7} ⩽ x ⩽ \frac{27 + 8 \sqrt{43}}{7}

\Rightarrow x_{m} = \frac{27}{7}

\Rightarrow t h e p o i n t s l i e o n a n e l l i p s e w i t h

c e n t e r a t (\frac{27}{7}, \frac{19}{7})

l e t x = u + \frac{27}{7}, y = v + \frac{19}{7}

{(u + \frac{27}{7})}^{2} - (v + \frac{54}{7}) (u + \frac{27}{7}) + (2 v - \frac{11}{7}) (v + \frac{19}{7}) - 30 = 0

\Rightarrow u^{2} - u v + 2 v^{2} - \frac{344}{7} = 0

l e t u = r \cos φ, v = r \sin φ

r^{2} \cos^{2} φ - r^{2} \cos φ \sin φ + 2 r^{2} \sin^{2} φ - \frac{344}{7} = 0

\Rightarrow r^{2} = \frac{688}{7 [3 - \sqrt{2} \cos (2 φ - \frac{π}{4})]}

\Rightarrow r = \frac{4}{7} \sqrt{\frac{301}{3 - \sqrt{2} \cos (2 φ - \frac{π}{4})}}

\Rightarrow a = r_{m a x} = \frac{4}{7} \sqrt{\frac{301}{3 - \sqrt{2}}} = \frac{4 \sqrt{129 + 43 \sqrt{2}}}{7}

\Rightarrow b = r_{m i n} = \frac{4}{7} \sqrt{\frac{301}{3 + \sqrt{2}}} = \frac{4 \sqrt{129 - 43 \sqrt{2}}}{7}

x = x_{m} + r \cos φ = \frac{27}{7} + \frac{4 \cos φ}{7} \sqrt{\frac{301}{3 - \sqrt{2} \cos (2 φ - \frac{π}{4})}}

y = y_{m} + r \sin φ = \frac{19}{7} + \frac{4 \sin φ}{7} \sqrt{\frac{301}{3 - \sqrt{2} \cos (2 φ - \frac{π}{4})}}

f = 2 x^{2} - 3 x + 4 y = f (φ)

\frac{d f}{d φ} = (4 x - 3) \frac{d x}{d φ} + 4 \frac{d y}{d φ} = 0

\dots \dots

\Rightarrow f_{m a x} \approx 242.6081

\Rightarrow f_{m i n} \approx - 11.4231

Commented by mr W last updated on 26/Jan/20

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