a、b都为正数,且a≠b,用不等号连接,a^5+b^5_______a^4×b+a×b^4
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![a、b都为正数,且a≠b,用不等号连接,a^5+b^5_______a^4×b+a×b^4](/uploads/image/z/3011217-33-7.jpg?t=a%E3%80%81b%E9%83%BD%E4%B8%BA%E6%AD%A3%E6%95%B0%2C%E4%B8%94a%E2%89%A0b%2C%E7%94%A8%E4%B8%8D%E7%AD%89%E5%8F%B7%E8%BF%9E%E6%8E%A5%2Ca%5E5%2Bb%5E5_______a%5E4%C3%97b%2Ba%C3%97b%5E4)
a、b都为正数,且a≠b,用不等号连接,a^5+b^5_______a^4×b+a×b^4
a、b都为正数,且a≠b,用不等号连接,a^5+b^5_______a^4×b+a×b^4
a、b都为正数,且a≠b,用不等号连接,a^5+b^5_______a^4×b+a×b^4
答案是>
作差:(a^5+b^5)-(a^4*b+a*b^4)
移项:=(a^5-a^4*b)+(b^5-a*b^5)
提公因式:=a^4(a-b)+b^4(b-a)
=(a^4-b^4)*(a-b)
=(a^2+b^2)(a^2-b^2)*(a-b)
=(a^2+b^2)(a+b)(a-b)*(a-b)
=(a^2+b^2)(a+b)(a-b)^2>0
(a^2+b^2)和(a-b)^2是肯定大于0的,而a和b又都是正数,所以整个(a^2+b^2)(a+b)(a-b)^2就大于0了,所以呢(a^5+b^5)>(a^4*b+a*b^4)了.
a^5+b^5>=a^4×b+a×b^4
(a^5+b^5)-(a^4×b+a×b^4)
=a^4(a-b)+b^4(b-a)
=(a-b)(a^4-b^4)
=(a-b)(a^2+b^2)(a^2-b^2)
=(a-b)(a^2+b^2)(a+b)(a-b)
=(a-b)^2(a^2+b^2)(a+b)>=0
因为a>0 b>0且a不等于b
所以(a*5+b*5)-(a*4b+b*4a)
=(a*5 -a*4b)-(b*4a-b*5)=a*4(a-b)-b*4(a-b) =(a*4-b*4)(a-b) =(a*2-b*2)(a*2+b*2)(a-b)
=(a+b) (a-b)(a*2+b*2) (a-b) =(a-b)*2(a*2+b*2)(a+b)>0 由上可知: a*5+b*5>a*4b+b*4a
>
a^5+b^5-a^4*b-a*b^4=a^4(a-b)+b^4(b-a)=(a^4-b^4)(a-b)
可知无论a>b 还是a(a^4-b^4)(a-b)>0
另外也可以由排序不等式证明
正序和>=乱序和