Expert: Paul Klarreich - 3/29/2008

Question
let h(x)=min{x,( x raised power two)}, for every real no of x then which of following are correct.1)h is continous for all x. 2)h is differentiable for all x. 3)differentiation of h(x)=1 for x more than 1. 4)h is not differentiable at two values of x.

Answer
Questioner:   mujeeb
Category:  Calculus
Private:  No
 
Subject:  limits,continuity and differentiability
Question:  let h(x)=min{x,( x raised power two)}, for every real no of x then which of following are correct.1)h is continous for all x. 2)h is differentiable for all x. 3)differentiation of h(x)=1 for x more than 1. 4)h is not differentiable at two values of x.
 
Hi, Mujeeb,

Your function is h(x)=  smaller of {x,x^2}

The only issue here is what happens at  x = +- 1, and what happens as  x -> +-1 from the left and right.  So you really are interested in:

h = either {x, x^2}, whichever you like.
As it happens, when -1<x<1, x^2 is the smaller, when x = +-1 they are equal, and  x is smaller outside of [-1,1]

1) Yes.  h(+-1) = 1 from either side.  h is cont at all x.

2) h'(x) = either  { 1, 2x }  As x -> +-1,  1 --> 1, but 2x -> +-2.  Since you don't get the same answers, h' does not exist at  x = +-1.

h is differentiable for all x except +-1.

I think that answers the other questions.