Extension applications for the definition of monotonicity of functions

There are many useful conclusions about the monotonicity of functions, and the understanding and flexible application can help us solve problems.

• Conclusion 1: Known functions \ (f (x) \),\ (g (x) \) monotonically increment (or subtract) on interval \ (d\) , then \ (f (x) =f (x) +g (x) \) in \ (d\) monotonically increasing (or decreasing);

Proof: either fetch \ (x_1<x_2\in d\), then by \ (f (x) \),\ (g (x) \) on \ (d\) monotonically increment,

Then \ (f (x_1) <f (x_2) \),\ (g (x_1) <g (x_2) \),

\ (F (x_1)-F (x_2) =f (x_1) +g (x_1)-[f (x_2) +g (x_2)]\)

\ (=f (x_1)-F (x_2) +g (x_1)-G (x_2) <0\),

That is, the function \ (F (x) =f (x) +g (x) \) is monotonically incremented on \ (d\) ;

In the same token, the function \ (f (x) \),\ (g (x) \) is monotonically decreasing on the interval \ (d\) , then \ (f (x) =f (x) +g (x) \) in \ (d\) on monotonically decreasing;

Simple application: For example \ (y=x\) on \ (r\) monotonically increment,\ (y=x^3\) on the \ (r\) monotonically increment,

Then \ (y=x+x^3\) is monotonically increasing on \ (r\) , which can help us understand and grasp the nature of more functions.

• Conclusion 2: The known function \ (f (x) \) is monotonically incrementing on the interval \ (d\) , and\ (g (x) \) is monotonically decreasing on the interval \ (d\) , then \ (f (x) =f (x)-g (x) \ ) is monotonically increasing on \ (d\) ;

Proof: Imitation on completion.

Simple application: such as \ (y=x\) on the \ ((0,+\infty) \) monotonically increment,\ (y=\cfrac{1}{x}\) in \ ((0,+\infty) \) On monotonically decreasing,

The function \ (y=x-\cfrac{1}{x}\) is monotonically incremented on the interval \ ((0,+\infty) \) .

• Conclusion 3: Known function \ (f (x) \),\ (g (x) \) monotonically incrementing on interval \ (d\) , and \ (f (x) >0\),\ (g (x) >0\) , then \ (H (x) =f (x) \cdot g (x) \) is monotonically incremented on \ (d\) ;

Proof: either fetch \ (x_1<x_2\in d\), then by \ (f (x) \),\ (g (x) \) on \ (d\) monotonically increment,

Then \ (f (x_1) <f (x_2) \),\ (g (x_1) <g (x_2) \),

i.e. \ (f (x_1)-F (x_2) <0\),\ (g (x_1)-G (x_2) <0\),

\ (H (x_1)-H (x_2) =f (x_1) \cdot g (x_1)-F (x_2) \cdot g (x_2) \)

\ (=f (x_1) \cdot g (x_1)-F (x_1) \cdot g (x_2)-[f (x_2) \cdot g (X_2)-F (x_1) \cdot g (x_2)]\)

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]-g (x_2) \cdot[f (x_2)-F (x_1)]\),

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]\),

Because \ (f (x_1)-F (x_2) <0\),\ (g (x_1)-G (x_2) <0\), and \ (f (x) >0\),\ (g (x) >0\) ,

Then the upper type \ (f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]<0\),

i.e. \ (H (x_1)-H (x_2) <0\)

That is, the function \ (H (x) =f (x) +g (x) \) is monotonically incremented on \ (d\) ;

Simple application: function \ (f (x) =x\),\ (g (x) =e^x\) monotonically increment on interval \ ((0,+\infty) \) , and \ (f (x) >0\),\ (g (x) >0\),

Then \ (H (x) =x\cdot e^x\) is monotonically incremented on \ ((0,+\infty) \) ;

• Conclusion 4: Known function \ (f (x) \),\ (g (x) \) monotonically decreasing on interval \ (d\) , and \ (f (x) >0\),\ (g (x) >0 \), then \ (H (x) =f (x) \cdot g (x) \) is monotonically decreasing on \ (d\) ;

Proof: either fetch \ (x_1<x_2\in d\), then by \ (f (x) \),\ (g (x) \) on \ (d\) monotonically decreasing,

Then \ (f (x_1) >f (x_2) \),\ (g (x_1) >g (x_2) \),

i.e. \ (f (x_1)-F (x_2) >0\),\ (g (x_1)-G (x_2) >0\),

\ (H (x_1)-H (x_2) =f (x_1) \cdot g (x_1)-F (x_2) \cdot g (x_2) \)

\ (=f (x_1) \cdot g (x_1)-F (x_1) \cdot g (x_2)-[f (x_2) \cdot g (X_2)-F (x_1) \cdot g (x_2)]\)

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]-g (x_2) \cdot[f (x_2)-F (x_1)]\),

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]\),

Because \ (f (x_1)-F (x_2) >0\),\ (g (x_1)-G (x_2) >0\), and \ (f (x) >0\),\ (g (x) >0\) ,

Then the upper type \ (f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]>0\),

i.e. \ (H (x_1)-H (x_2) >0\)

That is, the function \ (H (x) =f (x) +g (x) \) is monotonically decreasing on \ (d\) ;

• Conclusion 5: Known function \ (f (x) \),\ (g (x) \) monotonically decreasing on interval \ (d\) , and \ (f (x) <0\),\ (g (x) <0\) , then \ (H (x) =f (x) \cdot g (x) \) is monotonically incremented on \ (d\) ;

Proof: either fetch \ (x_1<x_2\in d\), then by \ (f (x) \),\ (g (x) \) on \ (d\) monotonically decreasing,

Then \ (f (x_1) >f (x_2) \),\ (g (x_1) >g (x_2) \),

i.e. \ (f (x_1)-F (x_2) >0\),\ (g (x_1)-G (x_2) >0\),

\ (H (x_1)-H (x_2) =f (x_1) \cdot g (x_1)-F (x_2) \cdot g (x_2) \)

\ (=f (x_1) \cdot g (x_1)-F (x_1) \cdot g (x_2)-[f (x_2) \cdot g (X_2)-F (x_1) \cdot g (x_2)]\)

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]-g (x_2) \cdot[f (x_2)-F (x_1)]\),

\ (=f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]\),

Because \ (f (x_1)-F (x_2) >0\),\ (g (x_1)-G (x_2) >0\), and \ (f (x) <0\),\ (g (x) <0\) ,

Then the upper type \ (f (x_1) \cdot [G (X_1)-G (x_2)]+g (x_2) \cdot [F (x_1)-F (x_2)]<0\),

i.e. \ (H (x_1)-H (x_2) <0\),

That is, the function \ (H (x) =f (x) +g (x) \) is monotonically incremented on \ (d\) ;

Extension applications for the definition of monotonicity of functions