Question

Find the values of m and b that make the following function differentiable.

f(x) = x^2 x=<2 (greater than or equal to symbol)
mx+b x>2

Answer 1

@Data_LG2 @jim_thompson5910

Answer 2

first what would you do to make it continuous?

Answer 3

plot it? @freckles

Answer 4

I was thinking make sure the left and right limit as x approaches 2 are the same

Answer 5

after you have that equation

Answer 6

you need to make sure the left derivative and right derivative as x approaches 2 is also the same

Answer 7

you will have a system of equations to solve

Answer 8

i don't understand

Answer 9

can u do it step by step w/ me

Answer 10

have you done left and right limits before?

Answer 11

yes

Answer 12

For a function to be continuous at x=c
we need:
1) f(c) to exist
2) lim x->c f(x) to exist
2a) for 2 to happen we would need the following:
\[\lim_{x \rightarrow c^-}f(x)=\lim_{x \rightarrow c^+}f(x)\]
3) lim x->c f(x)=f(c)

Answer 13

can you apply this to your problem?

Answer 14

im trying it rn

Answer 15

\[\lim_{x \rightarrow 2^-}f(x)=\lim_{x \rightarrow 2^+}f(x) \text{ will give you one equation } \\ \text{ then you will also have to consider smoothness } \\ \lim_{x \rightarrow 2-}f'(x)=\lim_{x \rightarrow 2^+}f'(x) \text{ is the second equation }\]

Answer 16

\[\lim_{x \rightarrow 2^-}x^2 =\lim_{x \rightarrow 2^+}(mx+b) \\ \lim_{x \rightarrow 2^-}(2x)=\lim_{x \rightarrow 2^+}(m)\]