Let f(x) = 24x2 if x≤ −1, m x3 + b if x>−1. If f is differentiable everywhere. Then m= and b=

Question

Let

f(x) =

24x2 if x≤ −1,

m x3 + b if x>−1.

If

f is differentiable everywhere. Then

m= and b=

Answer 1

We need f(x) to be continuous at x = -1, so 24(-1)² = m(-1)³ + b.

Therefore, b - m = 24.

Also, the one sided derivatives must be the same when x = -1. So, 48(-1) = 3m(-1)².

Consequently, m = -16 and b = 40.