The maximum value of the function of f(x)=(1+x)0.61+x0.6in the interval [0, 1] is

f′(x)=0.6(1+x)−0.41+x0.6−0.6x−0.4(1+x)0.61+x0.62        =0.61+x0.6−x−0.4(1+x)11+x0.62(1+x)0.4        =0.61+x0.6x0.4−(1+x)1+x0.62(1+x)0.4x0.4        =0.6x0.4−11+x0.62(1+x)0.4x0.4<0∀x∈(0,1)Hence, f(x) is decreasing. Thus, f(x)max=f(0)=1