Consider the function f(x)=3x3+4x2+11, and let c be a number in the interval [01]. For what values of k is there a c in this interval such that ?
3c^3 + 4c^2 + 11 = k
let c=0
k = 11
let c = 1
k = 3+4+11 = 18
All the terms of f(x) are positive,
and f'(x) = 9x^2 + 8x
so the graph will be increasing for all positive values of x
so for all values of k, such that
11 ≤ k ≤ 18, there will be a value of c between 0 and 1 such that f(x) = k