You must use the hint

Think about the minimum m and maximum M of the numbers H(0), H(1/2), H(1). Note the interval [m,M] is certainly in the range of H.

 You must consider the range of a continuous function f:[a,b]->R to be a closed and bounded interval [f(w_0),f(x_0)], where m=f(w_0) is the minimum value of f and M=f(x_0) is the maximum of f.