Question

Piece wise function?

Answer 1

\[If~ (x)= x^2+5~~~~ if~ x<2 \]
\[7x-5~~~if~~x \ge2\]
for all real numbers x, which of the following must be true?

Answer 2

i. f(x) is continuous everywhere.
ii. f(x) is differentiable everywhere.
iii. f(x) has a local maximum at x=2.

Answer 3

what does it mean to be differentiable?

Answer 4

it means the derivative exists

Answer 5

since the derivative is a limit, it means the limit exists

Answer 6

all you have to do for this is check at \(x=2\)

Answer 7

plug in 2 in to both expressions
if you get the same number it is continuous
if you do not, it is not

Answer 8

then take the derivative of both expressions and plug in 2
if you get the same number it is differentiable at 2
if you do not, it is no

Answer 9

it is continuous but it isn't differentiable.

Answer 10

ok i believe you

Answer 11

yeah plug in 2, get out 9

Answer 12

the derivative of the top one is \(2x\) and the bottom one is \(7\) so not differentiable at 2