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Câu hỏi: Cho hàm số $f\left( x \right)$ có đạo hàm trên $\mathbb{R}$ sao cho ${f}'\left( x \right)<0;\forall x>0.$ Hỏi mệnh đề nào dưới đây đúng?
A. $f\left( e \right)+f\left( \pi \right)=f\left( 3 \right)+f\left( 4 \right).$
B. $f\left( e \right)-f\left( \pi \right)\le 0.$
C. $f\left( e \right)+f\left( \pi \right)<2f\left( 2 \right).$
D. $f\left( 1 \right)+f\left( 2 \right)=2f\left( 3 \right).$
A. $f\left( e \right)+f\left( \pi \right)=f\left( 3 \right)+f\left( 4 \right).$
B. $f\left( e \right)-f\left( \pi \right)\le 0.$
C. $f\left( e \right)+f\left( \pi \right)<2f\left( 2 \right).$
D. $f\left( 1 \right)+f\left( 2 \right)=2f\left( 3 \right).$
Vì $f\left( x \right)<0;\forall x>0$ nên hàm số nghịch biến trên $\left( 0;+\infty \right)$ suy ra
$f\left( 1 \right)>f\left( 2 \right)>f\left( e \right)>f\left( 3 \right)>f\left( \pi \right)>f\left( 4 \right)$
Khi đó
+ $\left\{ \begin{aligned}
& f\left( e \right)>f\left( 3 \right) \\
& f\left( \pi \right)>f\left( 4 \right) \\
\end{aligned} \right.\Rightarrow f\left( e \right)+f\left( \pi \right)>f\left( 3 \right)+f\left( 4 \right)$ nên A sai
+ $f\left( e \right)>f\left( \pi \right)$ nên $f\left( e \right)-f\left( \pi \right)>0$ nên B sai
+ $\left\{ \begin{aligned}
& f\left( e \right)<f\left( 2 \right) \\
& f\left( \pi \right)<f\left( 2 \right) \\
\end{aligned} \right.\Rightarrow f\left( e \right)+f\left( \pi \right)<2f\left( 2 \right)$ nên C đúng
+ $\left\{ \begin{aligned}
& f\left( 1 \right)>f\left( 3 \right) \\
& f\left( 2 \right)>f\left( 3 \right) \\
\end{aligned} \right.\Rightarrow f\left( 1 \right)+f\left( 2 \right)>2f\left( 3 \right)$ nên D sai
$f\left( 1 \right)>f\left( 2 \right)>f\left( e \right)>f\left( 3 \right)>f\left( \pi \right)>f\left( 4 \right)$
Khi đó
+ $\left\{ \begin{aligned}
& f\left( e \right)>f\left( 3 \right) \\
& f\left( \pi \right)>f\left( 4 \right) \\
\end{aligned} \right.\Rightarrow f\left( e \right)+f\left( \pi \right)>f\left( 3 \right)+f\left( 4 \right)$ nên A sai
+ $f\left( e \right)>f\left( \pi \right)$ nên $f\left( e \right)-f\left( \pi \right)>0$ nên B sai
+ $\left\{ \begin{aligned}
& f\left( e \right)<f\left( 2 \right) \\
& f\left( \pi \right)<f\left( 2 \right) \\
\end{aligned} \right.\Rightarrow f\left( e \right)+f\left( \pi \right)<2f\left( 2 \right)$ nên C đúng
+ $\left\{ \begin{aligned}
& f\left( 1 \right)>f\left( 3 \right) \\
& f\left( 2 \right)>f\left( 3 \right) \\
\end{aligned} \right.\Rightarrow f\left( 1 \right)+f\left( 2 \right)>2f\left( 3 \right)$ nên D sai
| ${f}'\left( x \right)<0;\forall x\in \left( a;b \right)$ thì $f\left( x \right)$ nghịch biến trên $\left( a;b \right),$ khi đó $f\left( b \right)<f\left( a \right).$ |
Đáp án C.