Proving f(x) > e^x for a Monotonic Function with f'(x) > f(x) and f(0) = 1

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In summary, the conversation discusses proving that for a given function f from R to R with f'(x) greater than f(x) and f(0)=1 for all x, it can be shown that for every x greater than 0, f(x) is greater than e^x. The conversation also includes a suggested proof using the derivative of f(x)/e^x.
  • #1
talolard
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Homework Statement


Let f be a function from R to R such that [tex] f'(x)>f(x), f(0)=1 [/tex] for every x. prove that for every [tex] x>0 f(x)>e^x [/tex]

The Attempt at a Solution


From the given information we know that [tex] f'(0) \geq 0 [/tex] and so [tex] f'(x) \geq 0 [/tex]

define [tex] h(x) = f(x)-e^x [/tex]
so h(0) =0

[tex] h'(x)=f'(x) -e^x = lim_{h ->0 } \frac {f(x+h)-f(x)}{h} -e^x= lim_{h ->0 } \frac {f(x+h)-f(x) -he^x}{h}= lim_{h ->0 } \frac {f(x+h)-f(x) -0}{h} = f'(x) \geq 0 [/tex]

Then h(x) is montonous rising and so is never smaller then zero. thus [tex] f(x) \geq e^x [/tex]Is this correct?

Thanks
Tal
 
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  • #2
Well you seemed to have defined f(0) = 0, so that h(0) = -1...
 
  • #3
oops. I made a typo, that was supposed to be f(0)=1. Fixed in the original post.
Thanks
 
  • #4
OK cool. Even though you have the right idea, the second to last equality in your main derivation cannot be justified, since you are taking the limit of some term in the numerator of a quotient which is a serious violation of the limit laws (basically you need to know all limits in the the numerator and denominator exist before you can perform such operations).

My advice would be to consider g(x) = f(x)/e^x instead and show g is increasing. At no point should you need to resort to the definition of the derivative.
 
  • #5
That seems much simpler:
Just to make sure i got it right:
h(x) = [tex] \frac {f(x)}{e^x}, h(0)=1 [/tex]
[tex] h'(x)= \frac {f'(x)e^x -f(x)e^x}{e^(2x)} \geq 0 [/tex] because f'(x)>f(x)
Thanks
Tal
 
  • #6
That's fine. Notice the last inequality is in fact strict, which only helps you I guess.
 
  • #7
Thanks alot. Much apreciated
 

1. What is Calculus?

Calculus is a branch of mathematics that involves the study of rates of change and accumulation. It is used to model and analyze various natural phenomena, such as motion, growth, and decay.

2. What is a mathematical proof?

A mathematical proof is a rigorous, logical argument that uses established axioms, definitions, and previously proven theorems to demonstrate the truth of a mathematical statement.

3. How is Calculus used?

Calculus has many practical applications, including physics, engineering, economics, and statistics. It is used to solve problems involving optimization, rates of change, and area and volume calculations.

4. What is the difference between differential and integral Calculus?

Differential Calculus deals with the study of rates of change and slopes of curves, while integral Calculus focuses on the accumulation of quantities and finding the areas under curves.

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