# "First recall the fundamental results about the concavity of the" #
# "graph of a function" \ f(x) ":" #
# \qquad \quad \ f''(x) > 0 \quad => \quad "the graph of" \ \ f(x) \ \ "is concave up;"#
# \qquad \quad \ f''(x) < 0 \quad => \quad "the graph of" \ \ f(x) \ \ "is concave down."#
# "[I apologize -- with regard to concavity of the graph of a" #
# "function, I'm not sure I know the language: concave/convex." #
# "I am used to the language: (concave up)/(concave down)." #
# "I hope what I provide here can help you !!]" #
# "So, to answer questions about the concavity of the graph of a" #
# "function, we need to find its second derivative first." #
# "The function we are given is:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \qquad f(x) \ = \ 3 x^3 + 2 x^2 - x + 9. #
# \qquad \qquad :. \qquad \qquad \qquad \ f'(x) \ = \ 9 x^2 + 4 x - 1. #
# \qquad \qquad :. \qquad \qquad \quad \ f''(x) \ = \ 18 x+ 4. \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad \qquad (1) #
# "So, we want to find where:" #
# \qquad \qquad \qquad \qquad \qquad \qquad \ \ f''(x) > 0 \qquad "and" \qquad f''(x) < 0. #
# "So, using eqn. (1), we must find where:" #
# \qquad \qquad \qquad \qquad \qquad \ \ \ 18 x+ 4 > 0 \qquad "and" \qquad 18 x+ 4 < 0. #
# "So, we solve these inequalities. One way to do this is to do it" #
# "directly, as below:" #
# \qquad \qquad \qquad \qquad \qquad \ \ 18 x+ 4 > 0 \qquad "and" \qquad 18 x+ 4 < 0 #
# \qquad \qquad \qquad \qquad \qquad \qquad 18 x > -4 \qquad "and" \qquad 18 x < -4. #
# "As" \ \ 18 \ \ "is positive, we can divide through both sides of these" #
# "inequalities by" \ 18, "without changing the order of the" #
# "inequality sign:" #
# \qquad \qquad \qquad \qquad \qquad \quad x > - 4 / 18 \qquad "and" \qquad x < - 4 / 18 #
# \qquad \qquad \qquad \qquad \qquad \quad \quad x > - 2 / 9 \qquad "and" \qquad x < - 2 / 9. #
# "Thus, we have:" #
# x > - 2 / 9 \ \ => \ \ f''(x) > 0 \ => \ "graph of" \ f(x) \ "is concave up;" #
# x < - 2 / 9 \ \ => \ \ f''(x) > 0 \ => \ "graph of" \ f(x) \ "is concave down." #
# "Summarizing:" #
# \qquad \qquad \qquad \quad "graph of" \ f(x) \ "is concave up:" \qquad \ x > - 2 / 9 #
# \qquad \qquad \qquad \quad "graph of" \ f(x) \ "is concave down:" \qquad \ x < - 2 / 9 #
# "In interval notation:" #
# "graph of" \ f(x) \ "is concave up on the interval:" \qquad \quad ( - 2 / 9, + infty ) #
# "graph of" \ f(x) \ "is concave down on the interval:" \ ( - infty, - 2 / 9 ). #
# "This is the answer to our question." #
