Explanation Proof: Given information: The equation that relates the thickness (in millimeters) of the cortex of children of average intelligence with their ages (in year) is given by, A ( t ) = − 0.00005 t 3 − 0.000826 t 2 + 0.0153 t + 4.55 ( 5 ≤ t ≤ 19 ) (1) Where, t is the ages of children in years. Formula used : Power rule of differentiation , d d x ( x n ) = n x n − 1 . Theorem used: First derivative test: Equate the first derivative of an equation to 0; it gives the point of extrema (either maxima or minima ). Second derivative test: Put the point value of variable obtained by First derivative test in the second derivative of the equation, if: (a) The numeric value of second derivative comes positive, then it is a point of minima. (b) The numeric value of second derivative comes negative, then it is a point of maxima. Calculation : Use power rule to differentiate equation (1) with respect to t and obtain A ′ ( t ) . A ′ ( t ) = d d t ( − 0.00005 t 3 − 0.000826 t 2 + 0.0153 t + 4.55 ) = d d t ( − 0.00005 t 3 ) + d d t ( − 0.000826 t 2 ) + d d t ( 0.0153 t ) + d d t ( 4.55 ) = − 0.00015 t 2 − 0.001652 t + 0.0153 + 0 = − 0.00015 t 2 − 0.001652 t + 0.0153 (2) Equate above equation to 0, − 0.00015 t 2 − 0.001652 t + 0.0153 = 0 0.00015 t 2 + 0.001652 t − 0.0153 = 0 The general form of a quadratic equation is, a x 2 + b x + c = 0 . The solution of above equation is given by, x = − b ± b 2 − 4 a c 2 a (3) Comparing given equation with the general form to get the values of coefficients, The value of a is 0

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Chapter 4.4, Problem 68E

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The cortex of children with average intelligence reaches maximum thickness at age 6 years.

Solution Summary: The author explains how the first derivative of an equation relates the thickness of the cortex of children with average intelligence with their ages (in year).

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