The question asks
"If f(x)= x² + 10*sin(x), show that there is a number c such that f(c)=1000"
so its asking to solve for x when f(x)=1000 right?
Not exactly.
The problem is asking you to show a solution exists, but it does not ask you to find it.
Sometimes it is easy to show the existence of a solution but hard to actually find the solution. If you can find a solution, great-- that certainly proves its existence.