Considering the function
-x + x + 4 if x > -1
4 if x = -1
3 - x if x < -1
I was asked to show that the equation f(x) = -1 has a solution between 1 and 2. (Function was already determined to be continuous.)
For my solution:
f(x) = -(1)^3 + 1 + 4 = 4
f(x) = -(2)^3 + 2 + 4 = -2
Given that f(x) is a continuous function and f(2) < -1 < f(1) then by the intermediate value theorem there exists some number c such that f(x) = -1
Unfortuntely, I only recieved half credit for the answer. Is there something I missed? Was "f(2) < -1 < f(1)" the issue? Should I have manipulated my equations so that 0 would have been the center number?
(ex. f(x) -x^3 + x + 4 + 1 = 0 )
Any suggestions would be fantastic!
I can suggest two possible reasons (though I find neither reasonable here).
1) Maybe you were expected to show f was continuous or at least note it.
2) Maybe you were expected to find the actual value such f(c)=-1.