Answer
There is a real number $c$ such that $~f(c) = 1000$
Work Step by Step
Let $g(x) = x^2$
Let $h(x) = 10~sin~x$
These two functions are continuous on the interval $(-\infty, \infty)$
$f(x) = x^2+10~sin~x~~$ is also continuous on the interval $(-\infty, \infty)$, since it is the sum of two continuous functions.
Let $x = 0$:
$f(0) = (0)^2+10~sin~(0) = 0$
Let $x = 100~pi$:
$f(100~\pi) = (100~\pi)^2+10~sin~(100~\pi) = 10,000~\pi^2$
Note that: $0 \lt 1000 \lt 10,000~\pi^2$
Since $f(x)$ is a continuous function, there must be a real number $c$ where $0 \lt c \lt 100~\pi~~$ such that $~f(c) = 1000$
