What is the extreme f(p,q)=3p^3+4q^3-2p^2q-4p-5q ?

The extreme f(p,q)=3p^3+4q^3-2p^2q-4p-5q is

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extreme f(p,q)=3p^3+4q^3-2p^2q-4p-5q

extreme

Minimum \frac{2 ( 59 + 661 )}{235}, \frac{61 + 9 661}{2 470 59 + 661}, Maximum - \frac{2 ( 59 + 661 )}{235}, - \frac{61 + 9 661}{2 470 59 + 661}, Saddle \frac{2 ( 59 - 661 )}{235}, - \frac{9 661 - 61}{2 470 59 - 661}, Saddle - \frac{2 ( 59 - 661 )}{235}, \frac{9 661 - 61}{2 470 59 - 661}

Solution steps

Find the critical points:

\frac{2 ( 59 + 661 )}{235}, \frac{61 + 9 661}{2 470 59 + 661}, - \frac{2 ( 59 + 661 )}{235}, - \frac{61 + 9 661}{2 470 59 + 661}, \frac{2 ( 59 - 661 )}{235}, - \frac{9 661 - 61}{2 470 59 - 661}, - \frac{2 ( 59 - 661 )}{235}, \frac{9 661 - 61}{2 470 59 - 661}

\frac{\partial ^{2} f}{\partial p ^{2}} = 24 q

D (p, q) = 24 q (18 p - 4 q) - 16 p^{2}

Check the sign of at each critical point

Check critical point

\frac{2 ( 59 + 661 )}{235}, \frac{61 + 9 661}{2 470 59 + 661} :

Minimum

Check critical point

- \frac{2 ( 59 + 661 )}{235}, - \frac{61 + 9 661}{2 470 59 + 661} :

Maximum

Check critical point

\frac{2 ( 59 - 661 )}{235}, - \frac{9 661 - 61}{2 470 59 - 661} :

Saddle

Check critical point

- \frac{2 ( 59 - 661 )}{235}, \frac{9 661 - 61}{2 470 59 - 661} :

Saddle

Minimum \frac{2 ( 59 + 661 )}{235}, \frac{61 + 9 661}{2 470 59 + 661}, Maximum - \frac{2 ( 59 + 661 )}{235}, - \frac{61 + 9 661}{2 470 59 + 661}, Saddle \frac{2 ( 59 - 661 )}{235}, - \frac{9 661 - 61}{2 470 59 - 661}, Saddle - \frac{2 ( 59 - 661 )}{235}, \frac{9 661 - 61}{2 470 59 - 661}

What is the extreme f(p,q)=3p^3+4q^3-2p^2q-4p-5q ?
The extreme f(p,q)=3p^3+4q^3-2p^2q-4p-5q is