Is f(x) Increasing for Positive x and a in (0,1)?

The first derivative of this function is f'(x)=1-a^(x+1)-a^(x+2)+(x+1)*(x+2)*log(a)*(a^(x+1)-a^(x+2))+a^(2x+3). The problem is that it is difficult to compare log(a) with the powers of a in this equation. Some help is needed in proving that f(x) is increasing in positive x.
  • #1
lys111
2
0
I am supposed to show f(x) is increasing in x when x is real and positive.

f(x)= [(x+2)*(1-a^(x+1)] / [1-(x+2)*(1-a)*a^(x+1)-a^(x+2)]

a is any real in (0,1); x is real and positive

I have taken and first derivative of f(x):

f'(x)=1-a^(x+1)-a^(x+2)+(x+1)*(x+2)*log(a)*(a^(x+1)-a^(x+2))+a^(2x+3)

The problem is I cannot compare log(a) with the power of a. Can any of you genius help me with a proof as to showing f'(x) >0? Or maybe there is some other way to show f(x) is increasing in positive x?
 
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  • #2
lys111 said:
f(x)= [(x+2)*(1-a^(x+1)] / [1-(x+2)*(1-a)*a^(x+1)-a^(x+2)]
Please clarify the bracketing, preferably with LaTex. Is it
##f(x)=\frac{(x+2)*(1-a^{x+1})}{1-(x+2)*(1-a)*a^{x+1}-a^{x+2}}## ?
 
  • #3
Yes, it is. I am sorry about the typing/.

f(x)=[itex]\frac{(x+2)*(1-a^{x+1})}{1-(x+2)*(1-a)*a^{x+1}-a^{x+2}}[/itex]
 

FAQ: Is f(x) Increasing for Positive x and a in (0,1)?

What is an increasing function?

An increasing function is a mathematical function where the output (dependent variable) increases as the input (independent variable) increases. This means that as the value of the input variable increases, the value of the output variable also increases.

How do I prove that a function is increasing?

To prove that a function is increasing, you need to show that for any two input values, the output value of the function corresponding to the larger input value is always greater than or equal to the output value of the function corresponding to the smaller input value.

What is the difference between a strictly increasing and a weakly increasing function?

A strictly increasing function is a function where the output value increases strictly (without being equal) as the input value increases. On the other hand, a weakly increasing function is a function where the output value can be equal as the input value increases. In other words, a strictly increasing function will never have two equal output values for different input values, while a weakly increasing function may have.

What are some common techniques for proving a function is increasing?

Some common techniques for proving a function is increasing include using the definition of an increasing function, using the first derivative test, using the second derivative test, and using induction.

Can a function be increasing on one interval and decreasing on another?

Yes, it is possible for a function to be increasing on one interval and decreasing on another. This can happen when there is a point in the function where the slope changes from positive to negative or vice versa, called a critical point. At this point, the function may change from increasing to decreasing or vice versa.

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