Calculus Grade 12 University

The critical numbers of a function are the points at which the derivative is equal to 0 or is undefined. To find the critical numbers, we need to calculate the derivative of the given function, which is f'(x) = 24x² + 4x³. As the equation has no real roots, the only critical number is x = 0. In this case, the absolute maximum value of the function is 8(2)³ + (2)⁴ = 8(8) + (16) = 80 and the absolute minimum value of the function is 8(-8)³ + (-8)⁴ = -512. Both of these values are obtained at x = 2 and x = -8 respectively. Therefore, the critical numbers of the function y = 8x³ + x⁴ for x e [-8,2] are x = 0 and the absolute maximum value of the function is 80 and the absolute minimum value of the function is -512.