What is the extreme P(a,b)=a^3-b^3+a^2+ab+b^2 ?

The extreme P(a,b)=a^3-b^3+a^2+ab+b^2 is Minimum(0,0),Saddle(-1/3 , 1/3)

Popular >

extreme P(a,b)=a^3-b^3+a^2+ab+b^2

extreme points

P (a, b) = a^{3} - b^{3} + a^{2} + ab + b^{2}

Minimum (0, 0), Saddle (- \frac{1}{3}, \frac{1}{3})

Solution steps

Find the critical points:

(0, 0), (- \frac{1}{3}, \frac{1}{3})

\frac{\partial ^{2} f}{\partial a ^{2}} = - 6 b + 2

D (a, b) = 12 a - 36 ab + 3 - 12 b

Check the sign of at each critical point

Check critical point

(0, 0) :

Minimum

Check critical point

(- \frac{1}{3}, \frac{1}{3}) :

Saddle

Minimum (0, 0), Saddle (- \frac{1}{3}, \frac{1}{3})

f (x) = 8 x^{3} + 81 x^{2} - 42 x - 8

x^{\frac{5}{3}} - 5 x^{\frac{2}{3}}

f (x, y) = x^{3} + 3 x y^{2} - 15 x + y^{3} - 15 y

f (x) = x + \frac{36}{x}

f (x) = \frac{x}{x ^{2} - x + 9}, 0 \leq x \leq 9

What is the extreme P(a,b)=a^3-b^3+a^2+ab+b^2 ?
The extreme P(a,b)=a^3-b^3+a^2+ab+b^2 is Minimum(0,0),Saddle(-1/3 , 1/3)