for what values of a and c is the function f(x)=
ax^2, x is less than or equal to 2
x+c, x is greater than 2
differentiable for all real values of x?
i did the limit from each side and got the equation 4a= 2+c
but how do i make another equation? thanks!
Really all the question is asking is that you give a value for a and a value for c that make the function continuous. There are actually an infinite number of solutions to this problem, because the function will be continuous as long as c = ax^2. C is a function of a; you can choose any value for a.
Edit: Uh, oh. I had a problem like this a few weeks ago, and solved it the way I typed it up above. I got it right, but I now realize that was just a happy accident. Now I know better.
and x+ c are continuous and differentiable for all x. The only question is what happens at x= a. The limit of , as x approaches 2, is 4a whle the limit of x+ c, as x approaches 2, is 2+ c so you have 4a= 2+ c, as you say. The derivative of is 8ax and that has limit 16a as x approaches 2. The derivative of x+ c is 1 for all x so in order that the function be differentiable at x= 2 we must have 16a= 1. It should be easy to solve 16a= 1 for a and then 4a= 2+ c for c.
Note that the fact that if a function is differentiable at x= a, the derivative is not necessarily continuous there. However, it is always true that the derivative has the "intermediate value property" so that the two one sided limits, if they exist, must be the same.