MT【303】估计

(2016浙江填空压轴题)
已知实数$a,b,c$则 (     )
A.若$|a^2+b+c|+|a+b^2+c|\le1,$则$a^2+b^2+c^2<100$
B.若$|a^2+b+c|+|a+b^2-c|\le1,$则$a^2+b^2+c^2<100$
C.若$|a+b+c|+|a+b-c|\le1,$则$a^2+b^2+c^2<100$
D.若$|a^2+b+c|+|a+b^2-c|\le1,$则$a^2+b^2+c^2<100$

分析:利用排除法
A中令$c=-10,a=b,a^2+a-10=0$
B中令$c=0,b=-10,a^2=10$
C中令$c=0,a=10,b=-10$
故选D.D中$1\ge|a^2+b+c|+|a+b^2-c|\ge|a^2+a+b^2+b|=|(a+\dfrac{1}{2})^2+(b+\dfrac{1}{2})^2-\dfrac{1}{2}|$
易得$\dfrac{3}{2}\ge(a+\dfrac{1}{2})^2+(b+\dfrac{1}{2})^2\ge(a+\dfrac{1}{2})^2$故$a^2<4$同理$b^2<4$
$1\ge|a^2+b+c|+|a+b^2-c|\ge|a^2+b+c|$故$c^2<92$,得$a^2+b^2+c^2<100$

注:若$|a^2+ b + c| + |b^2 + a - c|\le1$, 则$a^2 + b^2 + c^2\le9.9032\cdots$是

$65536k^8 - 1327104k^7 + 8736256k^6 - 21760832k^5 + 18368665k^4$
$- 11528502k^3 + 9119692k^2 - 4451760k + 792768=0$

的最大实根.

注:
$a^2 + b^2 + c^2 < 7 + 4(a^2 + b + c)^2 + 4(b^2 + a - c)^2\le7 + 4[|a^2 + b + c| + |b^2 + a - c|]^2\le11.$

练习:已知$x,y\in R$(       )

A.若$|x-y^2|+|x^2+y|\le1$,则$(x+\dfrac{1}{2})^2+(y-\dfrac{1}{2})^2\le\dfrac{3}{2}$
B.若$|x-y^2|+|x^2-y|\le1$,则$(x-\dfrac{1}{2})^2+(y-\dfrac{1}{2})^2\le\dfrac{3}{2}$
C.若$|x+y^2|+|x^2-y|\le1$,则$(x+\dfrac{1}{2})^2+(y+\dfrac{1}{2})^2\le\dfrac{3}{2}$
D.若$|x+y^2|+|x^2+y|\le1$,则$(x-\dfrac{1}{2})^2+(y+\dfrac{1}{2})^2\le\dfrac{3}{2}$
分析:排除法,A中令 $x=\dfrac{1}{2},y=-\dfrac{1}{2}$
C中令 $x=\dfrac{1}{2},y=\dfrac{1}{2}$
D中令 $x=-\dfrac{1}{2},y=\dfrac{1}{2}$
故选B