Proving f(x) has one real root

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In summary, the conversation discusses using Roll's theorem to prove that a function has only one real root by contradiction. The picture provided contains the solution to the problem. The question asks for clarification on the logic behind disproving that f'(x) can never equal zero and how it proves that f(x) has only one real root. The proof states that because f'(x) is always greater than zero, there is only one real root. However, the questioner is confusing this statement with its converse, which is not always true.
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Miike012
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The problem asked to prove by contradiction that the function has only one real root. They used Roll'es theorem and proved that f ' (x) =/= 0 then they concluded that because f ' (x) =/= 0 then by contradiction the function can have atleast one real root...


Look at the picture it has the answer to the solution...

Question:
I don't understand the logic... I made up a graph where f ' (x) = 0 but yet f(x) still only has one real root... so how is disproving that f'(x) can never equal zero prove that f(x) has atleast only one real root when I obviously showed a scenario in which f'= 0 but f only has one root
 

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The proof is saying because f'(x)>0 for all x, there is only one real root. You're confusing that statement with its converse, "f(x) has only one real root; therefore, f'(x)>0," which isn't generally true.
 
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thank you
 

1. How can I prove that f(x) has one real root?

To prove that f(x) has one real root, you can use the Intermediate Value Theorem. This theorem states that if a continuous function f(x) takes on two values, a and b, at points a and b, respectively, then it must also take on every value in between a and b. If f(a) and f(b) have opposite signs, then there must be at least one root between a and b. If you can show that f(x) is continuous and that f(a) and f(b) have opposite signs, then you can prove that f(x) has at least one real root.

2. What does it mean for a function to have one real root?

A function having one real root means that there is only one value of x where the function equals zero. This value is also known as the x-intercept or the solution to the equation f(x) = 0. In other words, when you graph the function, it will only intersect the x-axis once.

3. Can a function have more than one real root?

Yes, a function can have more than one real root. This means that there is more than one value of x where the function equals zero. When graphed, the function will intersect the x-axis at multiple points.

4. Is it possible for a function to have no real roots?

Yes, it is possible for a function to have no real roots. This means that there is no value of x where the function equals zero. When graphed, the function will not intersect the x-axis at all.

5. What other methods can be used to prove that f(x) has one real root?

Other than the Intermediate Value Theorem, you can also use the Rational Root Theorem, Descartes' Rule of Signs, or the Fundamental Theorem of Algebra to prove that f(x) has one real root. These methods involve analyzing the coefficients and degree of the function to determine the number of real roots it has.

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