Can you solve this math pls?

Question

Suppose f(x) is continuous in [a,b] and let
z = (1/2) (f(x1)) + (f(x2))

where x1,x2 "element" [a, b] . Show that there exists a point c "element" [a, b] such that f(c) = z

Answer 1

Notice that z is half way between f(x1) and f(x2). Since f is continuous on [a,b], f is continuous on [x1,x2]. A continuous function takes on every value between any two values, so f takes on the value z. That is, there is a point c in [a,b] such that f(c) = z.