搜搜考试网若f(x)满足f(x)+2f(1/x)=5x 求f(4)的值
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若f(x)满足f(x)+2f(1/x)=5x 求f(4)的值
f(x)为二次函数 若3f(x+1)-2f(x-1)=4x^2-6x 求f(x)的解析式
f(x)为二次函数 若3f(x+1)-2f(x-1)=4x^2-6x 求f(x)的解析式
1.
f(x)+2f(1/x) =5x
f(1/x)+2f(x)=5/x
则
f(x)=10/(3x)-5x/3
f(4) = 5/6 - 20/3 = -35/6
2.
3f(x+1)-2f(x-1)=4x^2-6x
设f(x)=ax^2+bx+c
3f(x+1)-2f(x-1)=3(a(x^2+2x+1)+b(x+1)+c) - 2(a(x^2-2x+1)+b(x-1)+c) = ax^2+(10a+b)x+c+5b+a
=4x^2-6x
a=4,
10a+b=-6 => b = -46
c+5b+a=0 => c = 226
所以,f(x)=4x^2-46x+226
f(x)+2f(1/x) =5x
f(1/x)+2f(x)=5/x
则
f(x)=10/(3x)-5x/3
f(4) = 5/6 - 20/3 = -35/6
2.
3f(x+1)-2f(x-1)=4x^2-6x
设f(x)=ax^2+bx+c
3f(x+1)-2f(x-1)=3(a(x^2+2x+1)+b(x+1)+c) - 2(a(x^2-2x+1)+b(x-1)+c) = ax^2+(10a+b)x+c+5b+a
=4x^2-6x
a=4,
10a+b=-6 => b = -46
c+5b+a=0 => c = 226
所以,f(x)=4x^2-46x+226
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