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

Aug 14, 2017

Concave up

Explanation:

You have to find the 2nd derivative to determine this.
So, take the derivative, then do it again.

The first derivative is the derivative of the quotient of 2 functions:

$\frac{d}{\mathrm{dx}} f \frac{x}{g} \left(x\right) = \frac{f ' \left(x\right) g \left(x\right) - f \left(x\right) g ' \left(x\right)}{g {\left(x\right)}^{2}}$

Now, $\frac{d}{\mathrm{dx}} {\left(x - 2\right)}^{2}$ = $\frac{d}{\mathrm{dx}} \left({x}^{2} - 4 x + 4\right)$ = 2x - 4

And of course $\frac{d}{\mathrm{dx}} \left(x - 1\right)$ = 1

so, fitting these into our forumula for the derivative of the quotient of two functions, we have

$\frac{d}{\mathrm{dx}} f \frac{x}{g} \left(x\right) = \frac{\left(2 x - 4\right) \left(x - 1\right) - \left({x}^{2} - 4 x + 4\right)}{x - 1} ^ 2$

= $\frac{\left(2 {x}^{2} - 6 x + 4\right) - \left({x}^{2} - 4 x + 4\right)}{x - 1} ^ 2$

= $\frac{{x}^{2} - 2 x}{x - 1} ^ 2$

We can evaluate this at x = 2. The value is 0, so x = 2 is indeed a critical point. (Minima or maxima).

To find if it's a minima (function is concave up at this point) or maxima (concave down), we have to take the derivative again.

I used a little algebraic sleight of hand to simplify taking the second deriviate:

$\frac{{x}^{2} - 2 x}{x - 1} ^ 2 = \frac{{x}^{2} - 2 x + 1 - 1}{x - 1} ^ 2$

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

= $1 - \frac{1}{x - 1} ^ 2$

To find the derivative of this, note that the constant 1 in this equation drops out.

So $\frac{d}{\mathrm{dx}} \left(1 - \frac{1}{x - 1} ^ 2\right)$ = $\frac{d}{\mathrm{dx}} \left(- \frac{1}{x - 1} ^ 2\right)$

Rewrite this as:

$\frac{d}{\mathrm{dx}} \left(- {\left(x - 1\right)}^{-} 2\right)$

which is an application of the chain rule you should be able to do in your head:

= $2 {\left(x - 1\right)}^{-} 3$

or $\frac{2}{x - 1} ^ 3$

At a value of x=2, this evaluates to 2. Which, being > 0, indicates the curve is concave upward at this point.

You can double check this at:
https://www.desmos.com/calculator

...paste in:
\frac{\left(x\ -2\right)^2}{x-1}

so see the graph of the original function