Tham khảo:

refer

refer

\(f\left(x\right)+2f'\left(x\right)+f''\left(x\right)=x^3+2x^2\)

\(\Leftrightarrow f\left(x\right)+f'\left(x\right)+f'\left(x\right)+f''\left(x\right)=x^3+2x^2\)

\(\Leftrightarrow f\left(x\right)+f'\left(x\right)+\left[f\left(x\right)+f'\left(x\right)\right]'=x^3+2x^2\)

Đặt \(f\left(x\right)+f'\left(x\right)=u\left(x\right)\) ta được:

\(u\left(x\right)+u'\left(x\right)=x^3+2x^2\)

\(\Leftrightarrow e^x.u\left(x\right)+e^x.u'\left(x\right)=e^x\left(x^3+2x^2\right)\)

\(\Leftrightarrow\left[e^x.u\left(x\right)\right]'=e^x\left(x^3+2x^2\right)\)

\(\Rightarrow e^x.u\left(x\right)=\int e^x\left(x^3+2x^2\right)dx=e^x\left(x^3-x^2+2x-2\right)+C\)

\(\Leftrightarrow e^x\left[f\left(x\right)+f'\left(x\right)\right]=e^x\left(x^3-x^2+2x-2\right)+C\)

Thay \(x=0\) vào ta được \(2=-2+C\Rightarrow C=4\)

\(\Rightarrow e^x.f\left(x\right)+e^x.f'\left(x\right)=e^x\left(x^3-x^2+2x-2\right)+4\)

\(\Leftrightarrow\left[e^x.f\left(x\right)\right]'=e^x\left(x^3-x^2+2x-2\right)+4\)

\(\Rightarrow e^x.f\left(x\right)=\int\left[e^x\left(x^3-x^2+2x-2\right)+4\right]dx\)

\(\Rightarrow e^x.f\left(x\right)=e^x\left(x^3-4x^2+10x-12\right)+4x+C_1\)

Thay \(x=0\) vào ta được: \(1=-12+C_1\Rightarrow C_1=13\)

\(\Rightarrow e^x.f\left(x\right)=e^x\left(x^3-4x^2+10x-12\right)+4x+13\)

\(\Rightarrow f\left(x\right)=x^3-4x^2+10x-12+\frac{4x+13}{e^x}\)

\(\Rightarrow\int\limits^1_0f\left(x\right)dx=\int\limits^1_0\left(x^3-4x^2+10x-12\right)dx+\int\limits^1_0\left(4x+13\right).e^{-x}dx\)

Tích phân trước bạn tự tính, tích phân sau cũng đơn giản thôi:

Đặt \(\left\{{}\begin{matrix}u=4x+13\\dv=e^{-x}dx\end{matrix}\right.\) \(\Rightarrow\left\{{}\begin{matrix}du=4dx\\v=-e^{-x}\end{matrix}\right.\)

\(\Rightarrow I=-\left(4x+13\right).e^{-x}|^1_0+4\int\limits^1_0e^{-x}dx=\frac{-17}{e}+13-4.e^{-x}|^1_0=17-\frac{21}{e}\)

Casio cho kết quả tích phân trước là \(-\frac{97}{12}\)

Vậy \(\int\limits^1_0f\left(x\right)dx=\frac{107}{12}-\frac{21}{e}\)

=>2x^2+2x-3x-3+x^2+2x=3x^2+12x+12

=>12x+12=x-3

=>11x=-15

=>x=-15/11

\(f\left(3\right)+2f\left(\frac{1}{3}\right)=2.3=6\)=>\(2f\left(\frac{1}{3}\right)=6-f\left(3\right)\Leftrightarrow f\left(\frac{1}{3}\right)=3-\frac{f\left(3\right)}{2}\)

\(f\left(\frac{1}{3}\right)+2f\left(3\right)=2.\frac{1}{3}=\frac{2}{3}\)=> \(3-\frac{f\left(3\right)}{2}+2f\left(3\right)=\frac{2}{3}\Leftrightarrow\frac{3}{2}f\left(3\right)=\frac{2}{3}-3=-\frac{7}{3}\)

\(f\left(3\right)=-\frac{14}{9}\)

\(2f\left(x\right)-3f\left(\frac{1}{x}\right)=x^3\)

Thay \(x=2\) vào đẳng thức trên ta có : \(2f\left(2\right)-3f\left(\frac{1}{2}\right)=8\)

\(\Leftrightarrow2\left[2f\left(2\right)-3f\left(\frac{1}{2}\right)\right]=16\Leftrightarrow4f\left(2\right)-6f\left(\frac{1}{2}\right)=16\)(1)

Thay \(x=\frac{1}{2}\) vào đẳng thức trên ta có : \(2f\left(\frac{1}{2}\right)-3f\left(2\right)=\frac{1}{8}\)

\(\Leftrightarrow3\left[2f\left(\frac{1}{2}\right)-3f\left(2\right)\right]=\frac{3}{8}\Leftrightarrow6f\left(\frac{1}{2}\right)-9f\left(2\right)=\frac{3}{8}\)(2)

Lấy (1) cộng (2) ta được : \(4f\left(2\right)-9f\left(2\right)=16+\frac{3}{8}\Leftrightarrow-5f\left(2\right)=\frac{131}{8}\)

\(\Rightarrow f\left(2\right)=\frac{131}{8}:\left(-5\right)=-\frac{131}{40}\)

Bài này dễ mà

Xét x = 2

=> 2f(2) - 3f(1/2) = 8

Xét x = 1/2

=> 2f(1/2) - 3f(2) = 1/8

Đặt a = f(2), b = f(1/2)

Ta có hệ PT:

2a - 3b = 8

2b - 3a = 1/8

<=>

2a = 8 + 3b

16b - 24a = 1

<=>

2a = 8 + 3b

16b - 12(8 + 3b) = 1

<=>

2a = 8 + 3b

16b - 96 - 36b = 1

<=>

2a = 8 + 3b

20b = -97

<=>

a = -131/40

b = -97/20

Vậy f(2) = -131/40