First of all, let's find derivative of our function:
"(x+x^2-x^3)'=1+2x-3x^2=0"
"D=4-4*(-3)=4^2, \\\\ x_1= \\cfrac{-2-4}{-6}=1, x_2=\\cfrac{-2+4}{-6}=-\\cfrac{1}{3},"
"f'(1-\\epsilon)>0, f'(1+\\epsilon)<0\n\\implies x=1 -" local maximum, now let's prove that it's maximum value of the function when "x\\in[0,+\\infin): f(0)=0<f(1), f''(x)=2-6x \\implies \\forall x \\in [1, +\\infin): f''(x)<0," for this reason:
"f'(x)" is monotonically decreasing function "\\forall x \\in [1, +\\infin)" "\\implies \\forall x>1: f'(x)<0 \\implies f(x) -" monotonically decreasing function. "f(1)=f_{max}(x)" for "x\\in[0, +\\infin)"
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