$Find the real solution of the equation $x^{2}-10=x \sin{x}$.$

Question

$Find the real solution of the equation $x^{2}-10=x \sin{x}$.$

Does $x^{2}-10=x \sin{x}$ have a real solution? Solve the task without drawing a graph.
Explain the solution step by step, explain how to obtain the answer from the given information methodically, clearly, and coherently.

Calculus Trigonometry Precalculus

Egbonagba

22

Report

Answer

The answer is accepted.

Join Matchmaticians Affiliate Marketing Program to earn up to a 50% commission on every question that your affiliated users ask or answer.

Answer 1

Answers can only be viewed under the following conditions:

The questioner was satisfied with and accepted the answer, or
The answer was evaluated as being 100% correct by the judge.

View the answer

Define
\[f(x)=x^2-10-x\sin x.\]
Then
\[f(0)=0^2-10-0=-10<0\]
and
\[f(2\pi)=(2\pi)^2-10-2\pi \sin (2\pi)=4\pi^2-10>0.\]
Hence $f(0)<0$ and $f(2\pi)>0$. By the intermidiate value theorem, there exists $c$ in $(0,2\pi)$ such that
$f(c)=0$. Hence
\[0=f(c)=c^2-10-c\sin c \Rightarrow c^2-10=c\sin c.\]
Thus the given equation has a real solution.

Erdos

4.8K

$Find the real solution of the equation $x^{2}-10=x \sin{x}$.$

Answer

Related Questions

Search