Find the intervals on which the function is increasing, decreasing, concave up, or concave down and find any local extreme values and inflection points.

Question

So I know how to do the above, but not with piece wise functions. Someone help me out?

Y={3-x^2, x<0 and x^2+1, x>=0}
I found the 1st and 2nd derivative for both, but I cannot set them to zero or anything because the 2nd derivative is a constant; y''=2 and y''=-2. I could graph this but it says to use algebra/analytic methods.

Thanks

Answer 1

clearly y is discontinuous at x=0, since
lim(x->0-) = 3
lim(x->0+) = 1

y'=-2x for x<0
y'=2x for x>0

y" = -2 for x<0
y" = +2 for x>0

so, y is increasing everywhere, since y'>0 everywhere it is defined.

y is concave down for x<0
y is concave up for y>0

Geez - just sketch the graphs, and all will become clear.

A useful tool is desmos.com, which can do piecewise graphs:

https://www.desmos.com/calculator/cqhgzeu7sg

Answer 2

Got it. Thank you for clearing that up!