1. 10 points Consider the following function f(x) = ( e ^−1/x^2 if x ̸= 0 a if x = 0. Find a value of a that makes f differentiable on (−∞, +∞).

Question

1. 10 points Consider the following function f(x) = ( e ^−1/x^2 if x ̸= 0 a if x = 0. Find a value of a that makes f differentiable on (−∞, +∞). No credit will be awarded if l’Hospital’s rule is used at any point, and you must justify all your work.

Answer 1

I am not sure how convincing this answer is.

If f(x) =0 for x≠0 and exp(-1/x²) otherwise. then ....

For x<1, ln x>-1/x² ... I can't justify this algebraically but it is clear graphically.∗

x>exp(-1/x²)

The derivative of f(x) as defined above becomes

lim (as x->0) [exp(-1/x²)]/x which is 0...as it should be since f(x) has a minimum point at x=0.