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):}`

Lf' (−1) = a(−1) = −a

f' (−1) = 2b(−1) = −2b

Lf' (−1) = Rf' (−1)

−a = −2b

a = 2b

The function is differentiable so, the function continuity 

L.H.S. = R.H.S.

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

b − 3 = −a + b, a = 3

and b = 6