Question

If the function f(x) is differentiable and f(x)= {ax^3-6x; if x<=1 bx^2+4; x>1 , then a = ?

Answer 1

take the derivatives
plug in 1
set them equal
solve for a

Answer 2

a function f(x) is differentiable if

1) f(x) is continuous

AND

2) f ' (x) is continuous

Answer 3

\[f(x) = ax^3-6x;~~if~~x \le1~~~~
bx^2+4;~~~x>1~,then~~a=\]

Answer 4

although i am a bit confused because there is no a in the second one

Answer 5

\[f(x)=ax^3-6x\\
f'(x)=?\]

Answer 6

f'(x)= 3ax^2-6

Answer 7

ok and
\[f'(1)=3a-6\]

Answer 8

\[f(x)=bx^2+4\\
f'(x)=?\]

Answer 9

Is the function this?
\[\Large f(x) = \begin{cases} ax^3-6x \ \ \ \text{if } x \le 1 \\ bx^2+4 \ \ \ \ \ \ \text{if } x>1 \end{cases}\]

Answer 10

f'(x)= 2bx

Answer 11

right and
\[f'(1)=2b\]

Answer 12

yes, how did you do the piecewise like that?

Answer 13

set
\[3a-6=2b\] solve for \(a\) i guess

Answer 14

if you want to see the code, right click choose "show math as" then "latex"

Answer 15

well you need left limit=right limit of original function also

Answer 16

you will have a system of equations to solve

Answer 17

ooh right!!

Answer 18

yeah, freckles is correct, you need to make sure f(x) is continuous at x = 1

Answer 19

\[f(x) = \left\{\begin{array}{rcc}
ax^3-6x& \text{if} & x <1 \\
bx^2+4& \text{if} & x \geq 1
\end{array}
\right.\]
is another way to make a piecewise function

Answer 20

plug 1 in to both, set them equal

Answer 21

take the derivatives of both
plug in 1 again and set them equal

Answer 22

you will have two equations and two unknowns

Answer 23

satellite73 you have the inequality signs mixed up

Answer 24

the first one is supposed to be if x<=1 and the 2nd x>1

Answer 25

yeah i know

Answer 26

the beauty if it makes absolutely no difference

Answer 27

*is not "if"

Answer 28

ok so I pug in I get: a-6=b+4

Answer 29

you can almost do it in your head since 1 is so easy to compute with
the first equation is
\[a-6=b+4\]

Answer 30

yup

Answer 31

yeah i take the derivative

Answer 32

ok then plug in 1

Answer 33

take the derivative of \(ax^2-6a\) as before
you did this already

Answer 34

lol yeah but i made a typo
it is
\[ax^3-6x\]

Answer 35

3ax^2 - 6 and 2bx

Answer 36

yes

Answer 37

plug in 1, set them equal

Answer 38

3a-6=2b

Answer 39

k almost done

Answer 40

\[a-6=b+4\\
3a-6=2b\] solve that system

Answer 41

I got -14

Answer 42

now it is donkey work solving this
i wouldn't give it any attention, i would cheat
http://www.wolframalpha.com/input/?i=a-6%3Db%2B4%2C+3a-6%3D2b

Answer 43

\(a=-14\) is what wolfram got, so you must have done it right (i hope)

Answer 44

yay :) thanks

Answer 45

yw
and don't forget if you ever need code right click on it
good for copying and pasting too

Answer 46

how do I type a piece wise function?

Answer 47

Fanduekisses, right click on this code (shown below)

\[\Large f(x) = \begin{cases} ax^3-6x \ \ \ \text{if } x \le 1 \\ bx^2+4 \ \ \ \ \ \ \text{if } x>1 \end{cases}\]

go to "show math as" then go to "tex commands", you'll see something like this pop up

`\Large f(x) = \begin{cases} ax^3-6x \ \ \ \text{if } x \le 1 \\ bx^2+4 \ \ \ \ \ \ \text{if } x>1 \end{cases}`

you would type that out and surround it with `\[ \]` to have the server process your code and turn it into LaTex. You can also use the equation button below the text box