(a) y=x⁴-8x³-16x+5 has domain (-∞,∞), because there are no anomalies (like dividing by zero, for example).
To find the range we need to find dy/dx=4x³-24x²-16 and solve for dy/dx=0:
d²y/dx²=12x²-48x;
Newton’s Method using x₁=x₀-(4x₀³-24x₀²-16)/(12x₀²-48x₀), and subsequent iterations. Start with x₀=0, and after a few iterations we get x=6.107243 approx as the only real root of the cubic. Plug in this value of x and we get y=-523.867945 approx. Plug x into d²y/dx² and discover that it’s positive, implying a minimum at (6.107243,-523.867945). So the range is [-523.868,∞) approx.
(b) The y-intercept is 5. The x-intercepts require us to solve the quartic:
x⁴-8x³-16x+5=0.
Using Newton’s Method and starting first with x₀=0 we get x=0.299562.
With x₀=8, we get x=8.227394. Both are approximations. The other two solutions are complex.
(c) Concave up (minimum) for (6.107243,-523.867945).