Maxima and Minima

Question

1. For any real number x there is a unique integer n such that n≤x<n+1, and the greatest integer function is defined as ⌊x⌋=n. Where are the critical values of the greatest integer function? Which are local maxima and which are local minima?
2. Show that a cubic polynomial can have at most two critical points. Give examples to show that a cubic polynomial can have zero, one, or two critical points.

Answer 1

1.

I assume the definition

"A critical point is a value in the domain of the function where

the function is either not differentiable or the derivative is 0".

By this definition every real number is a critical point of the function ⌊x⌋.

And every point is also local minimum and local maximum.

2.

A cubic polynomial is always differentiable, so its critical points are

where its derivative is 0.

Its derivative is a quadratic polynomial.

Such polynomials have 0, 1, or 2 roots, which are then critical points of the cubic polynomial.

The following quadratic polynomials have 0, 1, 2 roots respectively:

x² + 1

x²

x² - 1

Therefore their integrals are cubic polynomials with 0, 1, 2 critical points:

x³/3 + x

x³/3

x³/3 - x