Question

cho hàm số y =f(x)=-5x

CMR: a. Nếu x\(_1\)<x\(_2\) thì f(x\(_1\))>f(x\(_2\))

b. f( x\(_1\)+ 4x\(_2\)) = f(x\(_1\))+4f(x\(_2\))

c. -f(x)=f(-x)

Answer 1

Lời giải:

Ta có:

\(f(x)=-5x\Rightarrow \left\{\begin{matrix} f(x_1)=-5x_1\\ f(x_2)=-5x_2\end{matrix}\right.\)

\(\Rightarrow f(x_1)-f(x_2)=-5x_1-(-5)x_2=-5(x_1-x_2)=5(x_2-x_1)\)

Do \(x_2> x_1\Rightarrow 5(x_2-x_1)>0\Leftrightarrow f(x_1)-f(x_2)>0 \)

\(\Leftrightarrow f(x_1)> f(x_2)\) (đpcm)

b)

\(\left\{\begin{matrix} f(x_1)=-5x_1\\ f(x_2)=-5x_2\rightarrow 4f(x_2)=-20x_2\end{matrix}\right.\)

\(\Rightarrow f(x_1)+4f(x_2)=-5x_1+(-20)x_2=-5x_1-20x_2\) (1)

Lại có:

\(f(x)=-5x\rightarrow f(x_1+4x_2)=-5(x_1+4x_2)=-5x_1-20x_2\) (2)

Từ (1),(2) suy ra \(f(x_1+4x_2)=f(x_1)+4f(x_2)\)

c)

\(f(x)=-5x\Rightarrow -f(x)=-(-5x)=5x\)

\(f(x)=-5x\Rightarrow f(-x)=-5(-x)=5x\)

Do đó: \(-f(x)=f(-x)\)