Plz some one help!!! Part a: Constract an infinitely differentiable function which vanishes outside a given finite interval [a,b]. Hint: Consider this function $$g(x) = e ^{-1/x ^{2}} $$ if x0 and 0 for x

Question

Plz some one help!!! Part a: Constract an infinitely differentiable function which vanishes outside a given finite interval [a,b]. Hint: Consider this function $$g(x) = e ^{-1/x ^{2}} $$ if x>0 and 0 for x<=0 Part b: Constract a infinitely differentiable function which is 0 forx<=a and 1 for x>=b (a

whimsical · Answer

i am not sure about what the question is trying to ask us to find, however my g'(x) is 
(2x^-3)(e^-x^-2)

anonymous · Answer

by saying the differentiable function vanishes outside a finite interval [a,b], you mean it doesn't exist for all x>b and x<a?

anonymous · Answer

I'll try my best to help so for part a.)
consider the square root function,
$$\sqrt{4-(x-2)^2}$$
Notice that if we graph this function, it vanishes outside the interval [0,4]...

now since the function is defined and is continous on the interval [0,4] or for 0<=x<=4,
then it is differentiable on any point on that interval, hence it must be infinitely differentiable function which vanishes outside the finite interval [0,4]

anonymous · Answer

thx a lot. Looks good.

anonymous · Answer

can I ask what you mean by an infinitely differentiable function?

anonymous · Answer

function that has derivatives of all orders

anonymous · Answer

ah.. So I could continue to differentiate it forever without ending up with zero right?

anonymous · Answer

right

anonymous · Answer

ok so my previous answer was wrong :))

anonymous · Answer

kindly disregard that :))

anonymous · Answer

why?

anonymous · Answer

your  function derivative never 0

anonymous · Answer

sorry carelessness :))

anonymous · Answer

any posiblilities to make it exponential function?

anonymous · Answer

yes, that's if the base is between 0 and 1