The number of pairs (a, b) of positive real n | vedclass

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(D) The given inequalities are $a^4+b^4  1$ for positive real numbers $a$ and $b$.Consider the curves $x^2+y^2=1$ (a circle of radi

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More than $2$

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(D) The given inequalities are $a^4+b^4  1$ for positive real numbers $a$ and $b$.Consider the curves $x^2+y^2=1$ (a circle of radius $1$) and $x^4+y^4=1$ (a squircle) in the first quadrant where $x, y > 0$.For any point $(x, y)$ on the circle $x^2+y^2=1$,we have $x^4+y^4 = (x^2+y^2)^2 - 2x^2y^2 = 1 - 2x^2y^2$. Since $x, y > 0$,$x^2y^2 > 0$,which implies $x^4+y^4 This means the region $a^4+b^4  1$ lies outside the circle. However,the curve $a^4+b^4=1$ actually encloses a larger area than the circle $a^2+b^2=1$ in the first quadrant.Specifically,for $a, b > 0$,the region $a^2+b^2 > 1$ and $a^4+b^4 Since this region has a non-zero area,there are infinitely many pairs $(a, b)$ of positive real numbers satisfying these inequalities.Thus,the number of such pairs is more than $2$.

The number of pairs $(a, b)$ of positive real numbers satisfying $a^4+b^4 < 1$ and $a^2+b^2 > 1$ is

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