Question

Cho hàm số f(x) có đạo hàm liên tục trên \(\left(\frac{1}{2};+\infty\right)\) và thoả mãn \(f\left(1\right)=-62\), \(f\left(x\right)< 0\), \(\left[-\left(2x+1\right)^4+2\left(2x+1\right)^2+1\right]f'\left(x\right)+\left(64x^3+96x^2+32x\right)f\left(x\right)=0\) với mọi \(x\in\left(\frac{1}{2};+\infty\right)\) . Khẳng định nào sau đây đúng

\(A.f\left(2\right)=-32\)

\(B.f\left(2\right)=-125\)

\(C.f\left(2\right)=-574\)

\(D.f\left(2\right)=-115\)

Answer 1

Đề đúng chứ bạn?

Biểu thức \(-\left(2x+1\right)^4+2\left(2x+1\right)^2+1\) chắc chắn đúng chứ?

Answer 2

\(\left[\left(2x+1\right)^4-2\left(2x+1\right)^2-1\right]f'\left(x\right)=\left(64x^3+96x^2+32x\right)f\left(x\right)\)

\(\Leftrightarrow\frac{f'\left(x\right)}{f\left(x\right)}=\frac{64x^3+96x^2+32x}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}=\frac{32x\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}\)

Lấy nguyên hàm 2 vế:

\(\int\frac{f'\left(x\right)}{f\left(x\right)}dx=\int\frac{32x\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}dx\)

\(\Leftrightarrow ln\left|f\left(x\right)\right|=\int\frac{32x\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}dx\)

Xét \(I=\int\frac{32x\left(x+1\right)\left(2x+1\right)}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}dx=\int\frac{4\left(2x+1-1\right)\left(2x+1+1\right)\left(2x+1\right)}{\left(2x+1\right)^4-2\left(2x+1\right)^2-1}d\left(2x+1\right)\)

Đặt \(t=2x+1\Rightarrow I=\int\frac{4\left(t-1\right)\left(t+1\right)t}{t^4-2t^2-1}dt=\int\frac{4t^3-4t}{t^4-2t^2-1}dt\)

\(=\int\frac{d\left(t^4-2t^2-1\right)}{t^4-2t^2-1}=ln\left|t^4-2t^2-1\right|+C\)

\(=ln\left|\left(2x+1\right)^4-2\left(2x+1\right)^2-1\right|+C\)

\(\Rightarrow ln\left|f\left(x\right)\right|=ln\left|\left(2x+1\right)^4-2\left(2x+1\right)^2-1\right|+C\)

Thay \(x=1\Rightarrow ln\left(62\right)=ln\left(62\right)+C\Rightarrow C=0\)

\(\Rightarrow ln\left|f\left(x\right)\right|=ln\left[\left(2x+1\right)^4-2\left(2x+1\right)^2-1\right]\)

\(\Rightarrow f\left(x\right)=-\left(2x+1\right)^4+2\left(2x+1\right)^2+1\)

\(f\left(2\right)=-574\)