How do you find what x-values are the graph on f(x)=2x−3x2 concave up and concave down?

Question

posciad48o · Accepted Answer

The graph of this function is concave up when

x > \frac{9}{2}

and concave down when

x < \frac{9}{2}

(the function is undefined at

x = 0

).
Explanation:
By the Quotient Rule, the first derivative is

f^{'} (x) = \frac{x^{2} \cdot 2 - (2 x - 3) \cdot 2 x}{x^{4}} = \frac{2 x - 4 x + 6}{x^{3}} = \frac{6 - 2 x}{x^{3}}

Using the Quotient Rule again gives the second derivative

f x) = \frac{x^{3} \cdot (- 2) - (6 - 2 x) \cdot 3 x^{2}}{x^{6}}

= \frac{- 2 x - 18 + 6 x}{x^{4}} = \frac{4 x - 18}{x^{4}}

Clearly

f x) > 0

when

x > \frac{9}{2}

and

f x) < 0

when

x < \frac{9}{2}

, making the graph of f concave up when

x > \frac{9}{2}

and concave down when

x < \frac{9}{2}

(and

x \neq 0

).

How do you find what x-values are the