What is the extreme f(s)=(4-s^2)/(2sx^2-7s-4) ?

The extreme f(s)=(4-s^2)/(2sx^2-7s-4) is Saddle(0,sqrt(7/2)),Saddle(0,-sqrt(7/2))

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extreme f(s)=(4-s^2)/(2sx^2-7s-4)

extreme points

f (s) = \frac{4 - s ^{2}}{2 s x ^{2} - 7 s - 4}

Saddle (0, \frac{7}{2}), Saddle (0, - \frac{7}{2})

Solution steps

Find the critical points:

(0, \frac{7}{2}), (0, - \frac{7}{2})

\frac{\partial ^{2} f}{\partial s ^{2}} = - \frac{4 s ( - 4 - 7 s - 6 s x ^{2} ) ( 4 - s ^{2} )}{( 2 s x ^{2} - 7 s - 4 ) ^{3}}

D (s, x) = \frac{384 s ^{7} x ^{8} - 3584 s ^{7} x ^{6} + 9408 s ^{7} x ^{4} - 19208 s ^{7} - 2304 s ^{6} x ^{6} + 12320 s ^{6} x ^{4} - 1568 s ^{6} x ^{2} - 46648 s ^{6} - 1536 s ^{5} x ^{8} + 14336 s ^{5} x ^{6} - 33152 s ^{5} x ^{4} - 3584 s ^{5} x ^{2} + 34496 s ^{5} + 7680 s ^{4} x ^{6} - 39424 s ^{4} x ^{4} - 8064 s ^{4} x ^{2} + 159488 s ^{4} - 13312 s ^{3} x ^{4} + 161024 s ^{3} + 2048 s ^{2} x ^{6} - 10752 s ^{2} x ^{4} + 4096 s ^{2} x ^{2} + 108416 s ^{2} - 2048 s x ^{4} + 33280 s - 4096 x ^{2}}{( - 4 - 7 s + 2 s x ^{2} ) ^{6}}

Check the sign of at each critical point

Check critical point

(0, \frac{7}{2}) :

Saddle

Check critical point

(0, - \frac{7}{2}) :

Saddle

Saddle (0, \frac{7}{2}), Saddle (0, - \frac{7}{2})

What is the extreme f(s)=(4-s^2)/(2sx^2-7s-4) ?
The extreme f(s)=(4-s^2)/(2sx^2-7s-4) is Saddle(0,sqrt(7/2)),Saddle(0,-sqrt(7/2))