What value(s) of c will make the following f(x) differentiable?Please give detailed explanation.?

Question

c E R: c is a real number

f:R->R

f is defined in 3 seperate parts

f(x)=4 * x<0.5> when x>0 I mean f is 4 times the square

root of x when x>0

f(x)=c when x=0

f(x)=4 * -x<0.5> when x<0 I mean f is 4 times the square root of

-x when x<0.

good luck

Answer 1

It cannot be made differentiable at x=0, no matter the value of c.

It&#039;s easy to see that f(x)→0 as x→0 from either direction, so for continuity we must have c=0. Continuity is necessary, but not sufficient, for differentiability. So I will take c=0 in what follows.

The one-sided limit

lim [f(x) - f(0)]/(x-0) is

x→0+

lim (4√x - 0)/x

x→0+

= lim 4/√x

x→0+

= +∞

However, if you consider the other one-sided limit

lim [f(x) - f(0)]/(x-0)

x→0-

you will find that this limit is -∞

The two one-sided limits don&#039;t agree, so there&#039;s no question that the two-sided limit, which is the definition of f&#039;(0), does not exist.