A critical point is defined to be any value in the domain were a function f is not differentiable or has the value 0. You have

f(x) = x^3-2x^2-5x+6 => f'(x) = 3x^2-4x-5

The function f(x) is differentiable everywhere. Thus we only have to search for the value in the domain were the derivative is 0, in the domain [4,8]

f'(x) = 0 <=> 3x^2-4x-5 = 0 <=> x = 2/3 +/- sqrt(19)/3

None of these values for x lies in the domain, hence the function doesn't have any critical points in the specified interval.