for which values of a and b will h(x) be differentiable at x= 0

Question

h(x) = e^(ax) for x≤ 0

       = √(x+b) for x>0

 

for which values of a and b will h(x) be differentiable at x= 0 ?

 

if it is differentiable, d/dx [e^(ax)] should equal d/dx [√(x+b)]

 

d/dx [e^(ax)] = e^(ax) • a

 

d/dx [√(x+b)] = 1/ [2√(x+b)]

 

e^(ax) • a = 1/ [2√(x+b)]

 

at x = 0

 

e^(a0) • a = 1/ [2√(0+b)]

a = 1/2√b

 

is this the right set up for the problem? where do i go from here?

Answer 1

The function should be continuous at x=0 and derivative should be the same at x=0.  You will create a system of equations.  One equation for equality at x=0, and another for same slope of tangent line at x=0.

 

Functions equal to each other.

 

e^a0 = √(0 + b)

 

e = √b

 

b = e²

 

 

Then we set the derivatives equal to each other.

 

ae^ax = (1/2)(x + b)^-1/2

 

when x=0,

 

a = (1/2)b^-1/2

 

a = 1 / (2√b)