Question

Consider the following piece-wise defined function.
Part I) If a=-3 and b=4, will f(x) be continuous at x=2?
Part II) If a=-3 and b=4, will f(x) be differentiable at x=2?
Part III) For what vales of a and b will f(x) be both continuous and differentiable at x=2?

Answer 1

ax^2+bx+2, x is less than and equal to 2
ax+b, x >2

Answer 2

Part 1) I sub a=-3 b=4 and x=2 into f(x) and determine -2=-2 so therefore f(x) is continuous at x=2
Part 2) I sub a=-3 b=4 and x=2 into f'(x) and determine that the limit of f'(x) DNE at x=2

Answer 3

or I should say the limit as x approaches 2 of f'(x) DNE

Answer 4

what? solve for a and b.

Answer 5

Part 3) is where I got stuck
I took f(x) and sub x=2 and solve it, now I think I should have taken the derivative of the piecewise and sub x=2 and then solve it

Answer 6

Yes part 3) wants the a and b value that will make it both continuous and differentiable

Answer 7

Okay, so you found f'(x)?

Answer 8

doing it right now

Answer 9

I think it is a= -(1/3)b

Answer 10

not sure that is correct

Answer 11

What did you get for f'(x)?

Answer 12

Hmm that's the same thing I'm coming up with.

Answer 13

2ax+b for x<2
a for x>2
then I set that equal to each other
2ax+b=a
2a(2)+b=a
4a+b=a
b=-3a
b/-3=a

Answer 14

so did I take the right approach
I know that differentiablity implies continuity
I at not sure I spelled that correctly

Answer 15

You can solve for a and b a little further though.
You should get a system of two equations by equating the left and right limits of f(x), and also equating the left and right limits of f'(x)

Answer 16

The second equation for a and b comes form f(2).

Answer 17

sub -b/3 into a.

Answer 18

This gives you an equation:\[\Large\rm \lim_{x\to2^-}ax^2+bx+2\quad =\quad \lim_{x\to2^+}ax+b\]And so does this:\[\Large\rm \lim_{x\to2^-}2ax+b\quad=\quad \lim_{x\to2^+}a\]

Answer 19

ok not sure what to do, I am getting a=0 b=0

Answer 20

I don't think that is correct

Answer 21

The system:
\[\Large\rm 4a+2b+2=2a+b\]\[\Large\rm 4a+b=a\]Hmm I came up with a=2, I haven't checked b yet..

Answer 22

sorry did not think to go back to f(x)

Answer 23

yes a=2 and b=-6

Answer 24

cool c:

Answer 25

Thanks :)