Proving f(x)=g(x) for Some x in [a,b]

  • Thread starter Chris(DE)
  • Start date
In summary: In this case, you can set h(x) = g(x) - f(x) and use the fact that h(a) < 0 and h(b) > 0. By the IVT, there must be some c in [a,b] such that h(c) = 0, which means g(c) - f(c) = 0, or g(c) = f(c). Therefore, there exists some x in [a,b] such that f(x) = g(x). In summary, to prove that f(x) = g(x) for some x in [a,b], we can use the fact that h(x) = g(x) - f(x) and show that h(c) = 0
  • #1
Chris(DE)
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Homework Statement



Suppose f and g are continuous on [a,b] and that f(a)<g(a), but f(b)>g(b). Prove that f(x)=g(x) for some x in [a,b]

Homework Equations



We are studying continuous functions and only have 3 theorems. IVT, Boundeness and the fact there is a max value for x.

The Attempt at a Solution



I am having trouble with this function for no other reason but I don't know how to state things.

First I drew out the problem and saw that f(x)=g(x) for some x.

This has to happen based on the IVT at some point the two graphs must cross. I also know that f(x)-g(x)= 0 My issue again is how to put this in a proof that would hold water.

I look at some value f(c) that is between f(a) and f(b). I can do this because of IVT. This would be a point that lies in [a,b] and would map to c. That same point should also be part of g since g is continuous on the same interval. How can I argue that g(c)=f(c).

Sorry for sounding dumb. I am having trouble just putting it in words of a valid proof.


Thanks
 
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  • #2
what can you tell me of the function h(x)=g(x)-f(x) in [a,b]?
 
  • #3
You were nearly at your solution!

A lot of tricks dealing with MVTs and similar problems involve using an auxiliary equation to get what you want.
 

1. How do you prove that f(x) and g(x) are equal for some values of x within a given interval?

To prove that f(x) and g(x) are equal for some values of x within a given interval [a,b], you need to show that the values of f(x) and g(x) are equal for all values of x within that interval. This can be done by substituting different values of x within the interval into both functions and showing that the resulting outputs are the same.

2. What is the importance of proving f(x)=g(x) for some x in [a,b]?

Proving that f(x)=g(x) for some x in [a,b] is important because it establishes an equality between two functions. This can be useful in various mathematical and scientific applications, such as simplifying complex equations or solving problems involving multiple variables.

3. What are some methods for proving f(x)=g(x) for some x in [a,b]?

There are various methods for proving that f(x)=g(x) for some x in [a,b]. Some common methods include direct substitution, algebraic manipulation, and using mathematical identities or properties. You can also use graphical methods, such as plotting both functions on the same graph and showing that they intersect at all points within the interval.

4. Can you provide an example of a proof for f(x)=g(x) for some x in [a,b]?

Yes, for example, let f(x)=x^2 and g(x)=2x within the interval [0,5]. To prove that f(x)=g(x) for some x in [0,5], we can substitute different values of x within the interval into both functions. For x=2, f(x)=4 and g(x)=4, therefore f(x)=g(x) for x=2. Similarly, for x=4, f(x)=16 and g(x)=8, proving that f(x)=g(x) for x=4. This can be repeated for all values of x within the interval, thus proving that f(x)=g(x) for some x in [0,5].

5. What are some common mistakes to avoid when proving f(x)=g(x) for some x in [a,b]?

One common mistake to avoid is assuming that the functions are equal without properly proving it. It is important to show that the values of f(x) and g(x) are equal for all values of x within the given interval. Another mistake is using incorrect mathematical operations or not following the correct order of operations. It is also important to double-check any substitutions made to ensure they are accurate. Additionally, be sure to clearly state your proof and any assumptions made.

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