document.write( "Question 366019: Please help me solve this equation:
\n" ); document.write( "For which values of t is the curve $\"+x=+t+%2Blnt+\"$ , $\"+y=+t-lnt+\"$ concave upward? \n" ); document.write( "

Algebra.Com's Answer #261095 by jsmallt9(3758) $\"\"$ $\"About$

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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.)

\n" ); document.write( " $\"dx%2Fdt+=+1+%2B+1%2Ft\"$
\n" ); document.write( "2nd derivative of x with respect to t = $\"%28-1%29%2Ft%5E2\"$

\n" ); document.write( " $\"dy%2Fdt+=+1+-+1%2Ft\"$
\n" ); document.write( "2nd derivative of y with respect to t = $\"1%2Ft%5E2\"$

\n" ); document.write( "The second derivative of y with repect to x would be the ratio of the two second derivatives above:
\n" ); document.write( "2nd derivative of y with respect to x = (2nd derivative of y with respect to t)/(2nd derivative of x with respect to t)
\n" ); document.write( "or
\n" ); document.write( "2nd derivative of y with respect to x = $\"%281%2Ft%5E2%29%2F%28%28-1%29%2Ft%5E2%29\"$
\n" ); document.write( "which simmplifies to:
\n" ); document.write( "2nd derivative of y with respect to x = -1

\n" ); document.write( "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.

\n" ); document.write( "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. \n" ); document.write( "

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