Cho hàm số f(x)>0 có đạo hàm liên tục trên \(\l...

C

- Hướng dẫn giải

\(\begin{aligned} &\text { Ta có } f^{\prime \prime}(x) \cdot f(x)+\left[\frac{f(x)}{\cos x}\right]^{2}=\left[f^{\prime}(x)\right]^{2} \Leftrightarrow \frac{f^{\prime \prime}(x) \cdot f(x)-\left[f^{\prime}(x)\right]^{2}}{f^{2}(x)}=-\frac{1}{\cos ^{2} x}\\ &\left[\frac{f^{\prime}(x)}{f(x)}\right]^{\prime}=-\frac{1}{\cos ^{2} x} \Rightarrow \frac{f^{\prime}(x)}{f(x)}=-\tan x+C .\\& \operatorname{Vì}\left\{\begin{array}{l} f^{\prime}(0)=0 \\ f(0)=1 \end{array}\right. \text { nên } C=0\\ &\text { Do đó } \frac{f^{\prime}(x)}{f(x)}=-\tan x . \text { Suy ra } \int_{0}^{\frac{\pi}{3}} \frac{d(f(x))}{f(x)}=\int_{0}^{\frac{\pi}{3}}-\tan x \cdot d x=\left.\int_{0}^{\frac{\pi}{3}} \frac{d(\cos x)}{\cos x} \Leftrightarrow \ln f(x)\right|_{0} ^{\frac{\pi}{3}}=\left.\ln \cos x\right|_{0} ^{\frac{\pi}{3}}\\ &\Leftrightarrow \ln f\left(\frac{\pi}{3}\right)-\ln f(0)=\ln \frac{1}{2}-\ln 1 \Leftrightarrow f\left(\frac{\pi}{3}\right)=\frac{1}{2} \end{aligned}\)