Question

been stuck on his one forever !

(a) find the critical numbers of (if any),
(b) find the open interval(s) on which the function is increasing
or decreasing, (c) apply the First Derivative Test to identify all
relative extrema, and (d) use a graphing utility to confirm your
results.

f(x)=x/x+1

Answer 1

\[\large f(x)=\frac{x}{x+1}\]
Let's add and subtract 1 on the top, in order to simplify before differentiating.\[\large f(x)=\frac{x+1-1}{x+1} \qquad =\qquad \frac{x+1}{x+1}-\frac{1}{x+1}\qquad = \qquad 1-\frac{1}{x+1}\]We'll write the second term with a negative exponent so it's easier to differentiate.\[\large f(x)=1-(x+1)^{-1}\]

Taking the derivative gives us,\[\large f'(x)=(x+1)^{-2} \qquad = \qquad \frac{1}{(x+1)^2}\]

Setting out derivative equal to 0 shows that we have a potential critical point at x=-1.
But at this point we need to remember back to the function we started with.
x=-1 is not in the DOMAIN of our function, so we don't care about that point.

Answer 2

So no critical points.. hmm

Answer 3

If we look at our derivative function, we can see that it will ALWAYS output positive values due to the square in the bottom.

This tells us that this function is ALWAYS increasing since it always has a positive slope.

Answer 4

thank you ![: