Proving Uniqueness of x in x^2 = xsinx + cosx Using Mean Value Theorem"

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In summary, the conversation discusses proving the equation x^2 = xsinx + cosx for two unique values of x in the set of real numbers. The speaker mentions using the Intermediate Value Theorem to prove the existence of at least one value, but is unsure how to prove the uniqueness. They mention using a geometric argument and ask for advice or ideas. The other speaker suggests noticing that the function is even.
  • #1
cochemuacos
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Mean value theroem??

Homework Statement



Show that [tex] x^2 = xsinx + cosx [/tex] is true only for two values of [tex] x \in {R} [/tex]

Homework Equations



Intermediate value theorem
Mean value theorem (?)

The Attempt at a Solution



I already know how to prove that there is al least one [tex] x \in [1,1.5] [/tex] and another [tex] x \in [-1.5,-1] [/tex] where the equation holds. The thing is that I'm not completely sure how to pove that they are unique, I have a geometric argument buy i feel it can be done using the mean value theorem.

Just for you to know, what i did to find out where the x's are, i took [tex] f(x) = x^2-xsinx-cosx [/tex] and gave values to the function it turns out that [tex] f(1) < 0 [/tex] and [tex] f(1.5) > 0 [/tex] so there must be at leat one [tex] x \in [1,1.5] [/tex] where [tex] f(x) = 0 [/tex] But that's it, I ran out of ideas although i feel I'm really close.

Any ideas or advices will be appreiciated
 
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  • #2


Have you noticed that the function is even? :smile:
 

FAQ: Proving Uniqueness of x in x^2 = xsinx + cosx Using Mean Value Theorem"

1. How does the Mean Value Theorem apply to proving uniqueness of x in x^2 = xsinx + cosx?

The Mean Value Theorem states that if a function is continuous on a closed interval and differentiable on the open interval, then there exists at least one point within the interval where the derivative of the function is equal to the average rate of change between the two endpoints. In the case of x^2 = xsinx + cosx, we can use the Mean Value Theorem to show that there is only one solution for x that satisfies the equation.

2. What is the general process for using the Mean Value Theorem to prove uniqueness of x in an equation?

The general process involves finding an appropriate function that satisfies the equation, showing that it is continuous and differentiable on a closed and open interval respectively, and then applying the Mean Value Theorem to show that there is only one solution for x within the interval.

3. Can the Mean Value Theorem be used for any equation involving trigonometric functions?

Yes, the Mean Value Theorem can be applied to any equation that involves continuous and differentiable functions, including trigonometric functions. It is a powerful tool for proving uniqueness of solutions in a variety of mathematical problems.

4. Are there any limitations to using the Mean Value Theorem for proving uniqueness of x in an equation?

While the Mean Value Theorem is a useful tool, it does have limitations. It can only be applied to equations with continuous and differentiable functions, and it may not always provide a definitive solution for uniqueness. In some cases, other mathematical techniques may be necessary to prove uniqueness.

5. Can the Mean Value Theorem be used to prove uniqueness of x for equations with more than one variable?

No, the Mean Value Theorem can only be applied to equations with one variable. For equations with more than one variable, other techniques such as partial derivatives or multivariable calculus may be necessary to prove uniqueness of solutions.

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