let f(x)$$\left[\begin{matrix}c^2\cos(x)-C & if x \le0 \ c^2x^3 & if 0

Question

let f(x)$$\left[\begin{matrix}c^2\cos(x)-C & if x \le0 \ c^2x^3 & if 0<x <1\ 3cx-2 & if 1\le x \end{matrix}\right]$$
Determine the value of "c" so that f(x) is continuous for all values of x. Thanks lots!

freckles · Answer

we will have to consider left limit of 0=right limit of 0
and left limit of 1=right limit of 1

freckles · Answer

$$c^2\cos(0)-C=c^2(0)^3 \ c^2(1)^3=3c(1)-2$$

anonymous · Answer

okay...

freckles · Answer

have you found C and c yet?

anonymous · Answer

no, not sure how to...

freckles · Answer

so you don't know how to simplify the equations I gave ?

freckles · Answer

cos(0)=1
0^3=0
1^3=1

freckles · Answer

$$c^2-C=0 \ c^2=3c-2$$

freckles · Answer

you have a system of equations to solve

freckles · Answer

find lower case c from the second equation
then you can go back in find big C from the first equation

freckles · Answer

don't be afraid of the second equation
it is just a quadratic

freckles · Answer

you solved quadratics before

freckles · Answer

I think

freckles · Answer

I assumed

freckles · Answer

$$Ax^2+Bx+C=0 \ x=\frac{-B \pm \sqrt{B^2-4AC}}{2A}$$

anonymous · Answer

does this work: (c-2)(c-1)

freckles · Answer

or factoring

freckles · Answer

(c-2)(c-1)=0
when c=? or c=?

anonymous · Answer

c=2,1

freckles · Answer

it just says to find little c

freckles · Answer

and there are two possible little c's

freckles · Answer

I wonder if they meant that big C to be a little C

anonymous · Answer

idk

anonymous · Answer

thanks for your help! appreciated it!

freckles · Answer

I want to add one more thing
since your question asks for little c and there is a C and a c
I kinda want to assume they meant C as c

freckles · Answer

and if so ...
$$c^2-C=0 \ \text{ is } c^2-c=0 \ c(c-1)=0 \ \text{ which means } c=0 or c=1 $$
but the intersection of both of these conclusions gives that c has to be 1

freckles · Answer

the thing about the question is weird because you cannot find c such that it makes the function continuous 
you have to find (c,C)

freckles · Answer

to make the function continuous but it doesn't say that

freckles · Answer

so that is why I'm drawing the conclusion there is a type-0

freckles · Answer

either in them asking to find the possible values (c,C)
or that they didn't mean to put a big C at all

freckles · Answer

you get what I'm saying @Brittni0605

freckles · Answer

wrong user

freckles · Answer

you get what I'm saying @esam2

anonymous · Answer

yes, i get what you mean.. i may need to check it with my teacher

freckles · Answer

we will do both cases:
(c,C)=(c,c^2)=(2,4) or (1,1)

---
or if they meant C=c then the answer is just c=1

anonymous · Answer

what happen to x=2?

freckles · Answer

x=2?

freckles · Answer

that wasn't a endpoint so we didn't need to consider it for making the function continuous since the piece that x=2 is a part of is continuous itself at x=2 from left and right

anonymous · Answer

oooh.... smart human!