"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."
