Question

Find all values of c such that f is continuous on (-[infinity], [infinity]). f(x) = { 3 - x^2 x less than or equal to c x, x > ca. c = -1 + squareroot 13/2, -1 - squareroot 13/2.b. c = 0. c. c = -1 = squareroot 13/2.d. c = 2.e. c = -1 + squareroot 13/2, 1 - squareroot 13/2.

Answer 1

a)
`c=(-1+√(13))/(2)` and
`c=(-1-√(13))/(2)`

Explanation:

The idea for the solution of this equation is to find the value of c where both parts of the piecewise-defined function are the same. So we need to take the parts of the function and set them equal to each other, so we get:

`3-x^(2)=x`

and then solve for x. We move everything to one side of the equation so we get:

`x^(2)+x-3=0`

and we use the quadratic formula:

`x=(-b\pm √(b^2-4ac))/(2a)`

and we substitute:

`x=(-1\pm √((1)^2-4(1)(-3)))/(2(1))`

and solve

`x=(-1\pm √(1+12))/(2)`

`x=(-1\pm √(13))/(2)`

so our two answers are:

a)
`c=(-1+√(13))/(2)` and
`c=(-1-√(13))/(2)`