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Wholesum continuity and differentiability (Posted on 2021-11-25) Difficulty: 3 of 5
Consider the following function

f(x)=(a+|x|)2e(5-|x|)2

i) Find all the values of a for which f(x) is continuous for all real x
ii) Find all the values of a for which f(x) is differentiable for all real x

No Solution Yet Submitted by Danish Ahmed Khan    
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Solution Solution Comment 1 of 1
The function is continuous for all real x for any choice of a.
(I tell my algebra students the domain of a function is all real numbers... unless it isn't.) 

The function is smooth everywhere except at x=0.  Think of the function without the absolute values around the x's.  f(x) is just the values with positive x reflected over the y-axis.  To make it smooth at x=0, we need the left and right hand derivatives of f(x) at x=0 to be equal.  By symmetry, they will both need to  equal zero.

I'll spare the details.
For the left replace |x| with -x, take the derivative and substitute x=0 to get -2a(5a-1)e^25
For the right replace |x| with x, take the derivative and substitute x=0 to get 2a(5a-1)e^25

These are both equal if a=0 or a=1/5

Answers
i) a is any real number
ii) a is 0 or 1/5






  Posted by Jer on 2021-11-27 09:40:22
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