Explanation d 2 y d x 2 is the second derivative of y with respect to x . c ( t ) = ( x ( t ) , y ( t ) ) is the parametric form of the curve y = f ( x ) where t is the parametric. If d 2 y d x 2 is positive, the curve is concave up. If d 2 y d x 2 is negative, the curve is concave down. Given: Let c ( t ) = ( t 2 , t 4 − 4 t ) Here, x ( t ) = t 2 , y ( t ) = t 4 − 4 t Formula used: d 2 y d x 2 = x ' ( t ) y ' ( t ) − y ' ( t ) x ' ' ( t ) ( x ' ( t ) ) 3 --------------------- (i) d d t ( t n ) = n t n − 1 --------------------------- (ii) d d t ( c o n s tan t ) = 0 -------------------- (iii) x ' ( t ) = d x ( t ) d t = d d t ( t 2 ) = 2 t 2 − 1 = 2 t ; [using (ii)] y ' ( t ) = d d t ( y ( t ) ) = d d t ( t 4 − 4 t ) = d d t ( t 4 ) − 4 d d t ( t ) = 4 t 4 − 1 − 4 ( 1 ) = 4 t 3 − 4 x ' ' ( t ) = d 2 d t 2 ( x ( t ) ) = d d t ( x ' ( t ) ) = d d t ( 2 t ) = 2 d d t ( t ) = 2 ( 1 ) = 2 y ' ' ( t ) = d d t ( y ' ( t ) ) = d d t ( 4 t 3 − 4 ) = 4 d d t ( t 3 ) − d d t ( 4 ) = 4

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Chapter 12.1, Problem 96E

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The $t$ -interval on which $c (t)$ is concave up.

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Chapter 12.1, Problem 95E

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