Consider the function f(x) = -2x^3 + x^2 -3x + 4
the intermediate value theorem says that if f is continuous on the interval [a,b] and k is in between f(a) and f(b), then there exists some value c, such that f(c) = k
in other words, if the function is continous on some interval, the function must pass through some value in between the values of the endpoints of the interval.
since the function is a polynomial, the function is continuous everywhere.
find f(-2) and f(4), the endpoints of the interval
f(-2) = 30
f(4) = -120
10 is in between 30 and -120, so there is some x value in between x=-2 and x=4 that makes the function value equal to 10.
to find that, set 10 equal to f(x)
\(10=-2x^3+x^2-3x+4\\ -2x^3+x^2-3x-6=0\\ x=-1\)
(i used a calculator to find x lmao)
so, when c = -1, f(c) = 10