Which of the following functions?

Question

Which of the following functions shows that the statement "If a function is continuous at x=0, then it is differentiable at x=0" is false?

 

a) f(x)=x^(-4/3)

b) f(x)=x^(-1/3)

c) f(x)=x^(1/3)

d) f(x)=x^(4/3)

e) f(x)=x^3

 

Answer: C

 

Please show all your work.

Answer 1

ƒ(x)=x^1/3

The general form of a derivative isƒ'(x)=n*x^n-1

Using this we get ƒ'(x)=(1/3)x^(1/3)-1→ƒ'(x)=(1/3)x^-2/3→ƒ'(x)=1/(3x^2/3)

 

When you plot this on a graph, you get a vertical asymptote at (0,0). A vertical asymptote represents dividing by 0, which is not possible, hence the function is not differentiable at (0,0)

 

 

 

Answer 2

Suppose I asked you the question:

Which of the following examples shows that the statement "If a pet dog is cool, then its owner is not cool" is false?

 

How would you answer? You would need an example of a pet dog which was cool but which also had a cool owner.

 

Similarly, you need an example of a function which is continuous at x=0 but *not differentiable* at x=0.

 

a) and b) are OUT because they are not continuous at x=0.  They both have vertical asymptotes at x=0.  That's cuz 0^(-4/3) = 1/(0^(4/3)) = 1/0 = undefined, same for 0^(-1/3) (because of the negative exponent).

 

Of c, d, and e, only 1 of them are *not differentiable* at x=0.  To see which one, take their derivatives, and set x=0.  Only one of them will not work because there'll be a 0 in the denominator.

 

Hope this helps!