In each of the following cases we want to find a function sa

In each of the following cases, we want to find a function satisfying various properties. For each part, is it possible to construct such a function? If you answer YES, give us an example, providing both an explicit equation for f and a sketch of its graph. If you answer NO, prove it. (a) A function f with the following properties f\' (1) exists f\" (1) does not exist.

Solution

We start backwards ie with f\'\'

LEt, f\'\'(x)=|x-1|

So f\'\'(x) exist at x=1 but, f\'\'\'(x) does nto exist at x=1

f\'(x)=(x-1)^2/2 , x>=1

f\'(x)=-(x-1)^2,2, x<1

f(x)=(x-1)^3/6, x>=1

f(x)=-(x-1)^3/6, x<1

PLot:

http://www.wolframalpha.com/input/?i=plot+y%3D%7Cx-1%7C%5E3%2F6

b)

g(x)=x^{1/2}

g is differentiable on (0,c) for all c >0

lim g at x=0+ =0

g\'(x)=x^{-1/2}/2

lim g\' at x=0+=\\infinity

Plot

http://www.wolframalpha.com/input/?i=plot+y%3Dx%5E(0.5),x%3E0

c)

Not possible

Because then at x=0 h(x) has to have a finite slope. But h(x) can\'t tend to infinity with a finite slope at x=0

It must tend to x=0 ie y axis asymptotically.