Find critical points and interval on which the function is increasing/decreasing

Question

f(x)= - (x² /x²-4) 

 

Find the critical points and intervals on which the function is increasing or decreasing . Use the first derivative test.

Answer 1

Presumably you know how to interpret the behavior of f' and so the main thing is to differentiate f correctly.

 

I prefer to use the product rule, differentiating f(x) = -x²(x²-4)^-1; let u = -x² and v = the other factor.

 

Then u' = -2x  and v' = -2x(x^2-4)^-2  and f' = u'v + uv' = 8x/(x²-4)².

 

The critical point is when f' = 0, which occurs if when x=0.  BTW, x ≠ ±2 (f & f' undefined)...so there are three critical points.

 

Here's the relevant numberlline showing CPs and the behaviors of f' and f on the four intervals created by the CPs:

 

.................(-2)............0...........2....................

f':    NEG      U    NEG   0   POS  U   POSITIVE      ⇐U means undefined

f:    decr.            decr.       incr.        incr.

 

There's a local minimum at x=0, where the function is continuous and switches from decreasing to increasing.

There are vertical asymptotes, x ±2.

The function is decreasing on (-infinity,-2) and (-2,0);

the function is increasing on (0,2) and (2,infinity).