Question

A⁴+b⁴ > или = a³b+ab³

Answer 1

$(a^4+b^4)-(a^3b+ab^3)=a^3(a-b)+b^3(b-a)=a^3(a-b)-b^3(a-b)=$

$=(a^3-b^3)(a-b)\ge 0,$ так как если a>b, то и a³>b³; если a<b, то и a³<b³ (ведь функция y=x³ монотонно возрастающая - чем больше аргумент, тем больше функция).

Answer 2

To compare the expressions $A^4 + B^4$ and $A^3B + AB^3$, where $A$ and $B$ are variables, you can simplify the expressions and analyze the inequality.

Let's simplify $A^4 + B^4$: $A^4 + B^4 = (A^2)^2 + (B^2)^2$ By the sum of squares formula $(x^2 + y^2)^2 = x^4 + 2x^2y^2 + y^4$, we can rewrite $A^4 + B^4$ as: $A^4 + B^4 = (A^2)^2 + 2(A^2)(B^2) + (B^2)^2 - 2(A^2)(B^2)$ $= (A^2 + B^2)^2 - 2(A^2)(B^2)$

And for $A^3B + AB^3$, it can be factored as: $A^3B + AB^3 = AB(A^2 + B^2)$

Now, comparing the two expressions: $A^4 + B^4 - 2(A^2)(B^2) \geq A^3B + AB^3$ $(A^2 + B^2)^2 - 2(A^2)(B^2) \geq AB(A^2 + B^2)$

To analyze this inequality, let's assume that $A$ and $B$ are positive real numbers. If $A^2 + B^2 = 0$, then the inequality is trivially true. Otherwise, divide both sides of the inequality by $A^2 + B^2$: $A^2 - 2AB + B^2 \geq AB$ $(A - B)^2 \geq AB$

For the inequality $(A - B)^2 \geq AB$ to be true, it is not possible to give a definite relationship between $A$ and $B$ without specific values for $A$ and $B$. The inequality holds for some values of $A$ and $B$ and does not hold for others, depending on the specific values of $A$ and $B$.

In conclusion, there isn't a simple general inequality relationship between $A^4 + B^4$ and $A^3B + AB^3$ without knowing specific numerical values for $A$ and $B$.