Question

For what values of a and b will be differentiable for all values of x? Discuss the geometry of the resulting graph of f.

Answer 1

where wer u stuck ?

Answer 2

basically stuck with how to approach this question haha

Answer 3

okie :)
First notice its a piecewise function

Answer 4

For a function to be differentiable,
1) It has to be continuous
2) slope must exist at all points

Answer 5

wid that knowledge, setup two equations and sovle \(a\) and \(b\)

Answer 6

i see, so wee need to check for continuity first then. \[\lim_{x \rightarrow 2^-}f(x)=\lim_{x \rightarrow 2^+}f(x)=4a-2b+3\]

Answer 7

f'(x) = 2ax - b = 4a-b = 0?

Answer 8

you need to set left/right limits equal for #1
you need to set the left/right slopes equal for #2

Answer 9

\(\lim_{x \rightarrow 2^-}f(x)= 2a \\ \\ \lim_{x \rightarrow 2^+}f(x)=4a-2b+3\)

Answer 10

set them equal

Answer 11

that takes care of continuity part

Answer 12

ah, i see, so 2a=4a-2b+3 -> 2a-2b+3=0
then f'(x)= a = f'(x)= 4a-b -> 3a-b=0 ?

Answer 13

Perfect !

Answer 14

okay thanks! now im able to find the values!(:

Answer 15

good, next visualizing the graph shoudl be easy

Answer 16

see if u can draw a rough sketch how it looks like...