Question

Find the values of the constants 𝑎, 𝑏 and 𝑐 for which the following function is differentiable. (Give 𝑎 and 𝑏 in terms of 𝑐.)

𝑓(𝑥)={𝑐𝑥^2+4𝑥+1, 𝑥≥1 & 𝑎𝑥+𝑏, 𝑥<1

I found a = 2c+4

b = ??? Seems like this is pretty easy, however I tend to overshoot problems so I can't figure this out, need some help, thank you!

Answer 1

for x < 1,
f(x) = ax+b
f'(x) = a

For x > 1,
f(x) = cx^2+4x+1
f'(x) = 2cx

So, as x -> 1 from the left
f(1) = a+b
f'(1) = a
As x -> 1 from the right
f(1) = c+5
f'(1) = 2c+4

So, we need
a+b = c+5
a = 2c+4

2c+4 + b = c+5
b = 1-c

So, for example, if c=3,
a = 10, b = -2
check:
f(x) = 3x^2 + 4x+1 for x > 1
f(x) = 10x-2 for x < 1
At x=1,
f(x) = 8, f'(x) = 10
for both sides.