SOLUTION: Please help me solve this equation: For which values of t is the curve {{{ x= t +lnt }}}, {{{ y= t-lnt }}} concave upward?

Question 366019: Please help me solve this equation:
For which values of t is the curve , concave upward?
Found 2 solutions by jsmallt9, robertb:
Answer by jsmallt9(3758) (Show Source): You can put this solution on YOUR website!
To determine the concavity we need the second derivative of y with respect to x. To find this we will find the second derivative of x with respect to t and the second derivative of y with respect to t. (Since the notation used in Calculus is not as standard as other parts of Math are and since Algebra.com's software does not make derivative notation easy, I am going to use more words than notation to explain what I'm doing.)

2nd derivative of x with respect to t =

2nd derivative of y with respect to t =

The second derivative of y with repect to x would be the ratio of the two second derivatives above:
2nd derivative of y with respect to x = (2nd derivative of y with respect to t)/(2nd derivative of x with respect to t)
or
2nd derivative of y with respect to x =
which simmplifies to:
2nd derivative of y with respect to x = -1

Since there is no variable in the 2nd derivative of y with respect to x, the concavity is a constant -1. In short, concavity is negative everywhere. This means the curve is concave downward everywhere.

So the answer to "For which values of t is the curve ... concave upward?" is: There are no values of t where the curve is concave upward.
Answer by robertb(5830) (Show Source): You can put this solution on YOUR website!
The answer given by the other tutor is wrong. He claims that the 2nd derivative of y with respect to x is the same as the 2nd derivative of y wrt t OVER the 2nd derivative of x wrt t. THAT IS NOT TRUE!
Now . Now , and . Hence . Hence
. (Used the quotient rule on the numerator derivative!)
For the parametric curve to be concave upward, , or . Solving this inequality, the critical numbers of the inequality are 0 and -1. Using the test numbers -2, -1/2, and 1, and checking for signs, the solution set is (, -1)U(0, ).
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