Need an explanation and answer

Question

How would you find all the minimum and maximum values of f(x)=x⁴-256? And would the maximum and minimum be relative or absolute?

Answer 1

You can notice that this function has a shape of an upward-facing parabola (just steeper) with the vertex at (0, -256). Thus the vertex is the absolute minimum. The smallest value x⁴ can have is zero, so the smallest value the function can have is -256. It is an absolute minimum because there are no other minima.

Alternatively, you can use calculus to find the first derivative and set it to zero (that gives you critical points such as max and min) 4x³=0. Which means x=0 is a critical point. Second derivative is equal to zero at this point so you cannot use it as a test to check if it is max or min value.

There is no maximum value for this function unless the problem also defines an interval where it should be considered.