# Is f(x)=1/(x-1)-2x concave or convex at x=0?

Apr 10, 2017

#### Answer:

$\text{concave at x = 0}$

#### Explanation:

To determine if f(x) is concave/convex at f( a) we require to find the value of f''( a)

• "If " f''(a)> 0" then f(x) is convex at x = a"

• "If " f''(a)<0" then f(x) is concave at x = a"

$f \left(x\right) = \frac{1}{x - 1} - 2 x = {\left(x - 1\right)}^{-} 1 - 2 x$

$\Rightarrow f ' \left(x\right) = - {\left(x - 1\right)}^{-} 2 - 2$

$\Rightarrow f ' ' \left(x\right) = 2 {\left(x - 1\right)}^{-} 3 = \frac{2}{x - 1} ^ 3$

$\Rightarrow f ' ' \left(0\right) = \frac{2}{- 1} ^ 3 = - 2$

$\text{Since " f''(0)<0" then f(x) is concave at x = 0}$
graph{(1/(x-1))-2x [-10, 10, -5, 5]}