What is the extreme xy^2+x^3y-xy ?

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extreme xy^2+x^3y-xy

extreme points

x y^{2} + x^{3} y - x y

Minimum (\frac{1}{5}, \frac{2}{5}), Maximum (- \frac{1}{5}, \frac{5 \frac{4}{5 5}}{2}), Saddle (1, 0), Saddle (- 1, 0)

Solution steps

Find the critical points:

(\frac{1}{5}, \frac{2}{5}), (- \frac{1}{5}, \frac{5 \frac{4}{5 5}}{2}), (1, 0), (- 1, 0)

\frac{\partial ^{2} f}{\partial x ^{2}} = 2 x

D (x, y) = - 9 x^{4} + 6 x^{2} - 4 y^{2} + 4 y - 1

Check the sign of at each critical point

Check critical point

(\frac{1}{5}, \frac{2}{5}) :

Minimum

Check critical point

(- \frac{1}{5}, \frac{5 \frac{4}{5 5}}{2}) :

Maximum

Check critical point

(1, 0) :

Saddle

Check critical point

(- 1, 0) :

Saddle

Minimum (\frac{1}{5}, \frac{2}{5}), Maximum (- \frac{1}{5}, \frac{5 \frac{4}{5 5}}{2}), Saddle (1, 0), Saddle (- 1, 0)

2 x^{3} - 27 x^{2} + 84 x, 1 \leq x \leq 8

f (x, y) = 1 0^{- 4} y - 2 \cdot 1 0^{- 8} y^{2}

f (x) = - 4 x^{2} + 4 x - 3

f (x) = x^{2} lo g_{10} (2 x)