有关函数的数学题求解答.（题目是英文的）.The polynomial f(x)=ax^4-2x^3+bx^2+2x has a factor of (x+1) and has a stationary value at x=1/2.Find the value of a and of b.
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有关函数的数学题求解答.（题目是英文的）.The polynomial f(x)=ax^4-2x^3+bx^2+2x has a factor of (x+1) and has a stationary value at x=1/2.Find the value of a and of b.
有关函数的数学题求解答.（题目是英文的）.
The polynomial f(x)=ax^4-2x^3+bx^2+2x has a factor of (x+1) and has a stationary value at x=1/2.Find the value of a and of b.
有关函数的数学题求解答.（题目是英文的）.The polynomial f(x)=ax^4-2x^3+bx^2+2x has a factor of (x+1) and has a stationary value at x=1/2.Find the value of a and of b.
应该可以用中文回答的吧,需要英文版追问我.
先求导f'(x)=4ax^3-6x^2+2bx+2
有因式（x+1）,即当x=-1,f(-1)=0=a+2+b-2=a+b (1)
当x=1/2时有极值,即f'(1/2)=0=4a*1/8-6*1/4+2b*1/2+2=a/2-3/2+b+2=0 (2)
由方程（1）（2）列方程组,即可解得a=1,b=-1
f '(x)=4ax^3-6x^2+2bx+2 f '(1/2)=1/2 *a-3/2+b+2=0 ①
f(x)=ax^4-2x^3+bx^2+2x has a factor of (x+1) , f(-1)=0=a+2+b-2 ②
1/2 *a-3/2+b+2=0 ①
0=a+2+b-2 ②
b=-1 ，a=1
f(x)=ax^4-2x^3+bx^2+2x=x(ax^3-2x^2+bx+2)
由题意可知(ax^3-2x^2+bx+2)|(x+1) (ax^3-2x^2+bx+2能被x+1整除)
不妨设ax^3-2x^2+bx+2=(x+1)(kx^2+mx+n)=kx^3+(k+m)x^2+(m+n)x+n
我们对比ax^3-2x^2+bx+2与kx^3+(k+m)x^2+(m+...
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f(x)=ax^4-2x^3+bx^2+2x=x(ax^3-2x^2+bx+2)
由题意可知(ax^3-2x^2+bx+2)|(x+1) (ax^3-2x^2+bx+2能被x+1整除)
不妨设ax^3-2x^2+bx+2=(x+1)(kx^2+mx+n)=kx^3+(k+m)x^2+(m+n)x+n
我们对比ax^3-2x^2+bx+2与kx^3+(k+m)x^2+(m+n)x+n
可知 k=a, k+m=-2, m+n=b, n=2
于是m=b-2 则k=a=-2-m=-2-(b-2)=-b 既 a+b=0
由f(x)在x=1/2处取的稳定点可知：
对于df(x)/dx=4ax^3-6x^2+2bx+2=0 ,x=1/2是它的一个解
于是将x=1/2带入方程，整理可得a+2b=-1
又因为a+b=0 于是解得a=1,b=-1
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