- #1
heman
- 361
- 0
How to Prove
[tex]x^2 = xsinx + cosx[/tex]
has exactly two real roots.
Will be thankful to yours bit of help.
[tex]x^2 = xsinx + cosx[/tex]
has exactly two real roots.
Will be thankful to yours bit of help.
shmoe said:With [tex]f(x)=x^{2}-x\sin x-\cos x [/tex], try using the intermediate value theorem to show you have at least one root in [itex](0,+\infty)[/itex], then use the fact that this is even (as dexter pointed out) to prove at least two roots.
Next consider what Rolle's theorem will say about the derivative of f(x) if you have more than two roots.
mathwonk said:how can it be that someone "would have never thought of" this, when it is a typical standard question and technique occurring in every cookbook calculus book in the united states?
saltydog said:Simple, I took Calculus 20 years ago. Doing it just for fun now
Ok, here is the solutionheman said:How to Prove
[tex]x^2 = xsinx + cosx[/tex]
has exactly two real roots.
Will be thankful to yours bit of help.
The purpose of proving this equation is to show the relationship between the functions x^2, xsinx, and cosx. This can help in solving other mathematical problems and understanding the behavior of these functions.
There are multiple approaches to proving this equation, but one common method is to use the properties of trigonometric identities and algebraic manipulation to simplify the left and right sides until they are equal.
Yes, this equation can be proven for all values of x since it is true for all real numbers. However, some values of x may require more complex methods or approximations to prove the equation.
The equation x^2 = xsinx + cosx can be applied in various fields such as physics, engineering, and economics. For example, it can be used to model the motion of a simple harmonic oscillator or to calculate the cost of production for a company.
Yes, there are alternative ways to prove this equation such as using geometric interpretations or using calculus concepts like derivatives and integrals. However, the basic principles of trigonometry and algebra will still be involved.