a simple method for comparing real functions

Let f(x) and g(x) be real-valued, twice differentiable functions on [a,b], and let x0 [a,b].

If f(x0)=g(x0), f(x0)=g(x0), f′′(x)g′′(x) for all x in [a,b], then f(x)g(x) for all x in [a,b].


Let h(x)=g(x)-f(x); by our hypotheses, h(x) is a twice differentiable function on [a,b], and by the Taylor formula with Lagrange form remainder (http://planetmath.org/RemainderVariousFormulas) one has for any x[a,b]:


where ξ=ξ(x)[x,x0].

Then by hypotheses,

h(x0) = g(x0)-f(x0)=0
h(x0) = g(x0)-f(x0)=0
h′′(ξ) = g′′(ξ)-f′′(ξ)0

so that


whence the thesis. ∎

Title a simple method for comparing real functions
Canonical name ASimpleMethodForComparingRealFunctions
Date of creation 2013-03-22 16:10:47
Last modified on 2013-03-22 16:10:47
Owner Andrea Ambrosio (7332)
Last modified by Andrea Ambrosio (7332)
Numerical id 10
Author Andrea Ambrosio (7332)
Entry type Result
Classification msc 60E15