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Lời giải:
Ta thấy: \(f(x)=\frac{x^3}{1-3x+3x^2}\Rightarrow f(1-x)=\frac{(1-x)^3}{1-3(1-x)+3(1-x)^2}=\frac{(1-x)^3}{3x^2-3x+1}\)
\(\Rightarrow f(x)+f(1-x)=\frac{x^3}{1-3x+3x^2}+\frac{(1-x)^3}{3x^2-3x+1}=\frac{x^3+(1-x)^3}{3x^2-3x+1}=1\)
Do đó:
\(f\left(\frac{1}{2017}\right)+f\left(\frac{2016}{2017}\right)=1\)
\(f\left(\frac{2}{2017}\right)+f\left(\frac{2015}{2017}\right)=1\)
............
\(f\left(\frac{1008}{2017}\right)+f\left(\frac{1009}{2017}\right)=1\)
Cộng theo vế:
\(\Rightarrow A=f\left(\frac{1}{2017}\right)+f\left(\frac{2}{2017}\right)+f\left(\frac{3}{2017}\right)+...f\left(\frac{2015}{2017}\right)+f\left(\frac{2016}{2017}\right)\)
\(=\underbrace{1+1+1...+1}_{1008}=1008\)
\(x^5+y^5-\left(x+y\right)^5\)
\(=x^5+y^5-\left(x^5+5x^4y+10x^3y^2+10x^2y^3+8xy^4+y^5\right)\)
\(=-5xy\left(x^3+2x^2y+2xy^2+y^3\right)\)
\(=-5xy\left[\left(x+y\right)\left(x^2-xy+y^2\right)+2xy\left(x+y\right)\right]\)
\(=-5xy\left(x+y\right)\left(x^2+xy+y^2\right)\)
Ta sẽ xét tính biến thiên của hàm số :
Ta có \(f\left(x\right)=\left(x^3-3x^2+3x-1\right)+4=\left(x-1\right)^3+4\)
\(f\left(\frac{2017}{2016}\right)-f\left(\frac{2016}{2015}\right)=\left(\frac{2017}{2016}-1\right)^3-\left(\frac{2016}{2015}-1\right)^3\)
\(=\left(\frac{1}{2016}-\frac{1}{2015}\right)\left[\left(\frac{2017}{2016}-1\right)^2+\left(\frac{2016}{2015}-1\right)^2+\left(\frac{2017}{2016}-1\right)\left(\frac{2016}{2015}-1\right)\right]\)
\(=\left(\frac{1}{2016}-\frac{1}{2015}\right)\left(\frac{1}{2016^2}+\frac{1}{2015^2}+\frac{1}{2016}.\frac{1}{2015}\right)< 0\)
\(\Rightarrow f\left(\frac{2017}{2016}\right)-f\left(\frac{2016}{2015}\right)< 0\Rightarrow f\left(\frac{2017}{2016}\right)< f\left(\frac{2016}{2015}\right)\)
Ta sẽ xét tính biến thiên của hàm số :
Ta có f\left(x\right)=\left(x^3-3x^2+3x-1\right)+4=\left(x-1\right)^3+4f(x)=(x3−3x2+3x−1)+4=(x−1)3+4
f\left(\frac{2017}{2016}\right)-f\left(\frac{2016}{2015}\right)=\left(\frac{2017}{2016}-1\right)^3-\left(\frac{2016}{2015}-1\right)^3f(20162017)−f(20152016)=(20162017−1)3−(20152016−1)3
=\left(\frac{1}{2016}-\frac{1}{2015}\right)\left[\left(\frac{2017}{2016}-1\right)^2+\left(\frac{2016}{2015}-1\right)^2+\left(\frac{2017}{2016}-1\right)\left(\frac{2016}{2015}-1\right)\right]=(20161−20151)[(20162017−1)2+(20152016−1)2+(20162017−1)(20152016−1)]
=\left(\frac{1}{2016}-\frac{1}{2015}\right)\left(\frac{1}{2016^2}+\frac{1}{2015^2}+\frac{1}{2016}.\frac{1}{2015}\right)< 0=(20161−20151)(201621+201521+20161.20151)<0
\Rightarrow f\left(\frac{2017}{2016}\right)-f\left(\frac{2016}{2015}\right)< 0\Rightarrow f\left(\frac{2017}{2016}\right)< f\left(\frac{2016}{2015}\right)⇒f(20162017)−f(20152016)<0⇒f(20162017)<f(20152016)
a.
\(x=9-\dfrac{1}{\sqrt{\dfrac{9-4\sqrt{5}}{4}}}+\dfrac{1}{\sqrt{\dfrac{9+4\sqrt{5}}{4}}}\\ x=9-\dfrac{1}{\dfrac{\sqrt{5}-2}{2}}+\dfrac{1}{\dfrac{\sqrt{5}+2}{2}}\\ x=9-\left(\dfrac{2}{\sqrt{5}-2}-\dfrac{2}{\sqrt{5}+2}\right)=9-8=1\\ \Rightarrow f\left(x\right)=f\left(1\right)=\left(1-1+1\right)^{2016}=1\)
c.
\(=\sin x\cdot\cos x+\dfrac{\sin^2x}{1+\dfrac{\cos x}{\sin x}}+\dfrac{\cos^2x}{1+\dfrac{\sin x}{\cos x}}\\ =\sin x\cdot\cos x+\dfrac{\sin^2x}{\dfrac{\sin x+\cos x}{\sin x}}+\dfrac{\cos^2x}{\dfrac{\sin x+\cos x}{\cos x}}\\ =\sin x\cdot\cos x+\dfrac{\sin^3x}{\sin x+\cos x}+\dfrac{\cos^3x}{\sin x+\cos x}\\ =\sin x\cdot\cos x+\dfrac{\left(\sin x+\cos x\right)\left(\sin^2x-\sin x\cdot\cos x+\cos^2x\right)}{\sin x+\cos x}\\ =\sin x\cdot\cos x-\sin x\cdot\cos x+\sin^2x+\cos^2x\\ =1\)
Ta có: \(\frac{1}{f\left(x\right)}-1=\frac{\left(1-x\right)^3}{x^3}\)
Xét hai số a, b dương sao cho \(a+b=1\)
Ta có: \(\hept{\begin{cases}\frac{1}{f\left(a\right)}-1=\frac{\left(1-a\right)^3}{a^3}\\\frac{1}{f\left(b\right)}-1=\frac{\left(1-b\right)^3}{b^3}\end{cases}}\)
\(\Rightarrow\hept{\begin{cases}\frac{1-f\left(a\right)}{f\left(a\right)}=\frac{\left(1-a\right)^3}{a^3}\\\frac{1-f\left(b\right)}{f\left(b\right)}=\frac{a^3}{\left(1-a\right)^3}\end{cases}}\)
\(\Rightarrow\frac{1-f\left(a\right)}{f\left(a\right)}.\frac{1-f\left(b\right)}{f\left(b\right)}=1\)
\(\Rightarrow f\left(a\right)+f\left(b\right)=1\)
Áp dụng vào bài toán ta được
\(f\left(\frac{1}{2017}\right)+f\left(\frac{2}{2017}\right)+...+f\left(\frac{2016}{2017}\right)\)
\(=\left[f\left(\frac{1}{2017}\right)+f\left(\frac{2016}{2017}\right)\right]+\left[f\left(\frac{2}{2017}\right)+f\left(\frac{2015}{2017}\right)\right]+...+\left[f\left(\frac{1008}{2017}\right)+f\left(\frac{1009}{2017}\right)\right]\)
\(=1+1+...+1=1008\)
Câu 2/
\(\hept{\begin{cases}2x^2-y^2+xy+3y=2\left(1\right)\\x^2-y^2=3\left(2\right)\end{cases}}\)
Ta có:
\(\left(1\right)\Leftrightarrow\left(x+y-1\right)\left(2x-y+2\right)=0\)
\(\Leftrightarrow\orbr{\begin{cases}y=1-x\\y=2x+2\end{cases}}\)
Thế ngược lại (1) giải tiếp sẽ ra nghiệm.
G(x)=x^2-2x+2
= x^2-x-x+1+1
=x(x-1)-1(x-1)+1
=(x-1)(x-1)+1
=(x-1)^2+1>=0+1>0
=) G(x) vô nghiệm
F(x)=3x(x-2)-x(3x+1)-14
=3x^2-6x-3x^2-x-14
=(3x^2-3x^2)+(-6x-x)-14
=-7x-14
Ta có: F(x)=-7x-14=0
(=) -7x=14
(=) x= -2
Vậy x=-2 là nghiệm của F(x)
a)\(\left(3x^2+x-2016\right)^2+4\left(x^2+506x-2017\right)^2=4\left(3x^2+x-2016\right)\cdot\left(x^2+506x-2017\right)\)
\(\Leftrightarrow\left(3x^2+x-2016\right)^2-4\left(3x^2+x-2016\right)\left(x^2+506x-2017\right)+4\left(x^2+506x-2017\right)^2=0\)
\(\Leftrightarrow\left(3x^2+x-2016-2x^2-1012x+4034\right)^2=0\)
\(\Leftrightarrow x^2-1011x+2018=0\)
\(\Leftrightarrow\left[{}\begin{matrix}x=2\\x=1009\end{matrix}\right.\)