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

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 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(3296)   (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(4012)   (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, ).