Question

Find the values of a and b so that the following function is differentiable for all values of x: 

f(x) = `{(ax + b","   x> -1),(bx^2 - 3","   x ≤ -1):}`

Answer 1

f(x) = `{(ax + b, x> -1), (bx^2 - 3,x ≤ -1):}` 

`f'(x) = {(ax, x > -1),(2bx, x≤ -1):}`

For f(x) to be differentiable at x = −1, it must be continuous there.

`lim_(x->-1^-) f(x) = b(-1)^2 - 3 = b - 3`

`lim_(x->-1^+) f(x) = ax + b = a(-1) + b = -a + b`

`lim_(x->-1^-) bx^2 - 3 = lim_(x->-1^+) ax + b`

b − 3 = −a + b

−3 = −a

a = 3

For x > −1: f(x) = ax + b = f'(x) = a = 3

For x < −1: f(x) = bx² − 3 = f'(x) = 2bx

At x = −1, f'(x) = 2b(−1) = −2b

3 = −2b

b = `-3/2`