SOLUTION: Use the method of contrapositive proof to prove the following statement.
Suppose x approaches R. If (x^3) -x is greater than 0, then x is greater than -1.
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Question 716373: Use the method of contrapositive proof to prove the following statement.
Suppose x approaches R. If (x^3) -x is greater than 0, then x is greater than -1.
Answer by solver91311(24713) (Show Source): You can put this solution on YOUR website!
The contrapositive of a statement has the same truth value as the statement. The contrapositive of 
is 
The interval of interest is then 
Take the first derivative of \ =\ x^3\ -\ x)
\ =\ 3x^2\ -\ 1)
Setting the first derivative equal to zero you can see readily that there are no critical points on the interval of interest.
The function is continuous and differentiable on the interval . Choose a test value in this interval, say -2.
Evaluate the first derivative at the test value:
\ =\ 3(-2)^2\ -\ 1\ =\ 11\ >\ 0)
Therefore the function is increasing on the interval, and therefore the value of the function at the right hand endpoint of the interval is the maximum value of the function on that interval.
\ =\ (-1)^3\ -\ (-1)\ =\ 0)
Since the maximum value of the function on the interval of interest is 0, the value of the function is less than or equal to zero whenever 
. Having proven the contrapositive true, the original statement is true.
Therefore: . Q.E.D.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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