Question

Calculus Vol. 1 Textbook, Question #96

Let m be the number of local minima and M be the number of local maxima. Can you create a function where M > m + 2? Draw a graph to support your explanation.

Answer 1

Solution::

Such a function is not possible if you want a continuous function because between two local minimum has to be a local maximum and vice versa so local minimum and maximum always alternate (if you exclude cases where f(x) is constant).

if you allow dis continuous function you can use

f(x) =[x]-x

it has local maximum at each integer  ∀x ∈ z

but no local minimum at all :for given x₀ ∈ r with [x]=k chose any

x ∈ (x₀ ,k-1):f(x₀) =k-x₀ so

> k-x=f(x) : x_{0 con't be local minimum}